{"id":63,"date":"2021-12-15T09:53:11","date_gmt":"2021-12-15T09:53:11","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/metrische-raeume\/"},"modified":"2021-12-15T09:53:11","modified_gmt":"2021-12-15T09:53:11","slug":"metrische-raeume","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/metrische-raeume\/","title":{"raw":"Metrische R\u00e4ume","rendered":"Metrische R\u00e4ume"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: 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4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z246d82a5a0f9\" class=\"sectionHead\"><span class=\"titlemark\">5.2 <\/span> <a id=\"x1-1390002\"><\/a>Metrische R\u00e4ume<\/h3> <a id=\"x1-139001r138\"><\/a> <h4 id=\"z1ba3e55997a7\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.1 <\/span> <a id=\"x1-1400001\"><\/a>Definition und erste Beispiele<\/h4> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z46668cc6c766\"> <a id=\"x1-140001r10\"><\/a> <span class=\"ecbx-1095\">Definition 5.10 <\/span>(Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Ein <span class=\"ecbx-1095\">metrischer Raum <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> ist eine Menge <math display=\"inline\"><mi>X<\/mi><\/math> gemeinsam mit einer Abbildung <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> die die <span class=\"ecbx-1095\">Metrik <\/span>auf <math display=\"inline\"><mi>X<\/mi><\/math> genannt wird und die folgenden drei Eigenschaften erf\u00fcllt: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Definitheit) F\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Symmetrie) F\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Dreiecksungleichung) F\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <p class=\"indent\">Intuitiv ausgedr\u00fcckt weist eine Metrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> auf einer Menge <math display=\"inline\"><mi>X<\/mi><\/math> je zwei Punkten ihre <span class=\"ecbx-1095\">Distanz <\/span>(ihren <span class=\"ecbx-1095\">Abstand<\/span>) zu. In dieser Auffassung besagt die Definitheit der Metrik, dass der einzige Punkt, der Abstand Null zu einem gegebenen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> hat, <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> selbst ist. Symmetrie der Metrik besagt, dass der Abstand von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> zu <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> der gleiche ist wie von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> zu <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Fasst man die Distanz zwischen zwei Punkten als die L\u00e4nge eines k\u00fcrzesten Weges vom einen zum anderen Punkt auf (was nicht immer m\u00f6glich ist), dann                                                                                                                                                                           besagt die Dreiecksungleichung, dass die L\u00e4nge eines k\u00fcrzesten Weges von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> nach <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/math> h\u00f6chstens so gross ist wie die L\u00e4nge eines Weges, den man abl\u00e4uft, wenn man zuerst den Umweg nach <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> und von dort aus nach <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/math> geht. <\/p><p class=\"indent\">Folgende Beispiele von Metriken sind uns eigentlich bereits bekannt \u2013 siehe Lemma <a href=\"..\/..\/chapter\/metrische-raeume#x1-140002r11\">5.11<\/a> unten: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math> mit der Standardmetrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> definiert durch <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math> mit der Standardmetrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> definiert durch <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> mit der Einsmetrik <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> definiert durch <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn> <\/mrow>  <mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> mit der euklidischen Metrik <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> definiert durch <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn> <\/mrow>  <mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> mit der Maximumsmetrik <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> definiert durch <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn> <\/mrow>  <mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> und damit auch <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> werden wir im Normalfall die euklidische Metrik <math display=\"inline\"><msub><mrow><mi>d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> ben\u00fctzen und diese auch einfach mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> bezeichnen.                                                                                                                                                                           <\/p><p class=\"indent\">Wie wir auch sehen werden, gibt es viele weitere, interessante Beispiele von metrischen R\u00e4umen. Manche aber nicht alle dieser erhalten wir mittels Normen <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><\/math> auf Vektorr\u00e4umen wie in Definition&nbsp;<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-136001r1\">5.1<\/a>. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z2787e27b433e\"> <a id=\"x1-140002r11\"><\/a> <span class=\"ecbx-1095\">Lemma 5.11 <\/span>(Eine Norm definiert eine Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">ein Vektorraum <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Hier und auch im Folgenden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir ebenso Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">betrachten,<\/span> <span class=\"ecti-1095\">doch inkludiert der Fall der reellen Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume auch den Fall von komplexen Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen, weshalb wir<\/span> <span class=\"ecti-1095\">Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">hier und im Folgenden nicht mehr getrennt erw<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnen werden.<\/span><\/span><\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><\/math> <span class=\"ecti-1095\">eine Norm<\/span> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann definiert<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Metrik <\/span><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>V<\/mi> <\/math><span class=\"ecti-1095\">, die man auch die<\/span> <span class=\"ecti-1095\">von der Norm <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> <span class=\"ecbi-1095\">induzierte Metrik <\/span><span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">nennt.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Es gilt f\u00fcr <math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach Definitheit der Norm <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Homogenit\u00e4t der Norm f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> gilt f\u00fcr <math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und somit erhalten wir die Symmetrie von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Zuletzt verwenden wir die Dreiecksungleichung der Norm und erhalten                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Dies zeigt die Dreiecksungleichung f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> womit also <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><\/math> eine Metrik auf <math display=\"inline\"><mi>V<\/mi> <\/math> ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Nicht jede Metrik auf einem Vektorraum muss durch eine Norm gegeben sein. Des Weiteren ist das Messen von Distanzen nicht nur auf Vektorr\u00e4umen von Interesse. Interessante Beispiele dieser Art m\u00f6chten wir nun besprechen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9d35aa01af6c\"> <a id=\"x1-140003r12\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.12 <\/span>(Weitere metrische R\u00e4ume)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Diskrete Metriken) Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge und <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer<\/span> <span class=\"ecti-1095\">Raum. In der Tat ist <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definit und symmetrisch per Definition. Des Weiteren erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">die Dreiecksungleichung:<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">Punkte<\/span> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gilt, dann ist<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">trivialerweise<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt. Falls <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist mindestens von<\/span> <span class=\"ecti-1095\">einem Punkt in <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">verschieden und die Dreiecksungleichung gilt ebenso.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Man beachte, dass die diskrete Metrik auf<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">nicht durch eine Norm gegeben ist. In der Tat w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde eine Norm<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">widerspr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chlicherweise die Homogenit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tseigenschaft in Definition <\/span><a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-136001r1\"><span class=\"ecti-1095\">5.1<\/span><\/a> <span class=\"ecti-1095\">nicht erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Manhattanmetrik) Wir setzen <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo> <\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">. In der Tat<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">alle Axiome<\/span> <span class=\"ecti-1095\">einer Metrik auf <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">, da<\/span> <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>X<\/mi><mo class=\"MathClass-bin\">\u00d7<\/mo><mi>X<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">die Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung<\/span> <span class=\"ecti-1095\">der Einsmetrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist. Die<\/span> <span class=\"ecti-1095\">Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">wird oft auch Manhattan-Metrik genannt. Grund daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ist, dass man in<\/span> <span class=\"ecti-1095\">schachbrettartig angelegten Orten wie zum Beispiel Manhattan auf folgende Weise von<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gelangt: Man geht zuerst bei<\/span> <span class=\"ecti-1095\">gleichbleibender <\/span><math display=\"inline\"><mi>y<\/mi><\/math><span class=\"ecti-1095\">-Koordinate<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und dann bei<\/span> <span class=\"ecti-1095\">gleichbleibender <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Koordinate<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">, oder<\/span> <span class=\"ecti-1095\">umgekehrt von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">dann von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Es g<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">be zwar noch andere M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten, aber wenn alle Strassen<\/span> <span class=\"ecti-1095\">in Manhattan von West-Ost oder Nord-S<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">d verlaufen, dann misst<\/span> <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">den relevanten Abstand zwischen zwei Punkten.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Metrik der franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Eisenbahn) Wir setzen<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und definieren<\/span> <span class=\"ecti-1095\">die SNCF-Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">SNCF<\/mi><mo>  <\/mo> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">SNCF<\/mi><mo>  <\/mo> <\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;linear&nbsp;abh\u00e4ngig&nbsp;\u00fcber&nbsp;<\/mtext><\/mstyle><mi>\u211d<\/mi><mstyle class=\"text\"><mtext>&nbsp;sind<\/mtext><\/mstyle> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;linear&nbsp;unabh\u00e4ngig&nbsp;\u00fcber&nbsp;<\/mtext><\/mstyle><mi>\u211d<\/mi><mstyle class=\"text\"><mtext>&nbsp;sind<\/mtext><\/mstyle><\/mtd><\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">. Der<\/span> <span class=\"ecti-1095\">Grund f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Namen dieser Metrik (siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140008r13\"><span class=\"ecti-1095\">5.13<\/span><\/a><span class=\"ecti-1095\">) ist, dass eine Bahnreise von einer franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Stadt bei<\/span> <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">zu einer anderen bei<\/span> <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">meist <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber den Ursprung<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">(auch Paris genannt)<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt, ausser wenn <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">auf derselben \u2013 von Paris ausgehenden geraden Strecke liegen. Gewissermassen besteht<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">in dieser Metrik also aus unendlich vielen Halbgeraden, die sich nur im Ursprung<\/span> <span class=\"ecti-1095\">treffen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Ein kombinatorischer Graph ist eine endliche Menge von Punkten, die sogenannten Ecken, von<\/span> <span class=\"ecti-1095\">welchen einige mit sogenannten Kanten verbunden sind. Diese lassen sich auf nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche<\/span> <span class=\"ecti-1095\">Weise mit mehreren Metriken ausstatten; der Konkretheit halber betrachten wir<\/span> <span class=\"ecti-1095\">einen spezifischen Graphen, doch muss ein Graph nicht unbedingt als Teilmenge<\/span> <span class=\"ecti-1095\">von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">gegeben sein.<\/span> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=86&amp;url=Pictures\/metrik\/graph.pdf\"><img id=\"z6f5e09d592f2\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/metrik\/graph.svg\" width=\"86\"><\/div>  <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">Man kann nun eine Metrik auf den Ecken (durch<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u2219<\/mo><\/math> <span class=\"ecti-1095\">gekennzeichnet) dadurch definieren, dass man benachbarten Ecken die Distanz<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">zuweist und dies iteriert. Beispielsweise definiert man die Distanz zweier<\/span> <span class=\"ecti-1095\">Ecken, die man <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber zwei aber nicht weniger Kanten erreichen kann,<\/span> <span class=\"ecti-1095\">als<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dazu notwendig ist, dass man von einer Ecke zu jeder anderen Ecke <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber Ablaufen von<\/span> <span class=\"ecti-1095\">Kanten gelangen kann (wie bei obigem Graphen) \u2013 diese Eigenschaft nennt sich auch<\/span> <span class=\"ecti-1095\">Zusammenhang des Graphen.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren ist es auch m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glich, eine Metrik auf dem kompletten<\/span> <span class=\"ecti-1095\">(kontinuierlichen) Graphen zu definieren, indem man die obige Definition auf<\/span> <span class=\"ecti-1095\">folgende Weise erweitert. Fasst man eine Kante als Kopie des Intervalles<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">auf,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">die zwei Ecken der Kante sind, so kann man eine Distanz auf den Kanten <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Distanz<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">definieren. <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">hnlich wie oben kann man nun damit eine Metrik auf dem gesamten Graphen<\/span> <span class=\"ecti-1095\">(inklusive den Kanten) definieren.<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zba227d1040eb\"> <a id=\"x1-140008r13\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.13.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die in Beispiel <\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140003r12\"><span class=\"ecti-1095\">5.12<\/span><\/a><span class=\"ecti-1095\">(iii) definierten Metriken tats<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chlich Metriken sind.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren Sie des Weiteren die Konstruktion der Metriken in (iv) vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndig und formal durch.<\/span> <\/p> <\/div> <p class=\"indent\">Wir bemerken, dass f\u00fcr eine gegebene Teilmenge <math display=\"inline\"><mi>Y<\/mi> <\/math> eines metrischen Raumes <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> die Einschr\u00e4nkung <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>Y<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo><mi>Y<\/mi> <\/mrow> <\/msub> <\/math> eine Metrik auf <math display=\"inline\"><mi>Y<\/mi> <\/math> definiert. Wenn <math display=\"inline\"><mi>Y<\/mi> <\/math> mit dieser Metrik versehen ist, nennen wir <math display=\"inline\"><mi>Y<\/mi> <\/math> einen <span class=\"ecbx-1095\">Teilraum<\/span> des metrischen Raumes <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und die Metrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>Y<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo><mi>Y<\/mi> <\/mrow><\/msub><\/math> die <span class=\"ecbx-1095\">induzierte Metrik<\/span>. Wenn nicht anders spezifiziert, statten wir Teilmengen eines metrischen Raumes implizit mit der induzierten Metrik aus. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z8b0a4c9e6b0e\"> <a id=\"x1-140009r14\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.14 <\/span>(Umgekehrte Dreiecksungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer Raum.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7b232840b8d1\"> <a id=\"x1-140010r15\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.15 <\/span>(Deformation der Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer Raum. Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt><mstyle class=\"mbox\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mover accent=\"true\"><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Metriken <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>d<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mover accent=\"true\"><mrow><mi>d<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">definieren.<\/span> <\/p> <\/div> <a id=\"x1-140011r140\"><\/a> <h4 id=\"zfab30b961798\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.2 <\/span> <a id=\"x1-1410002\"><\/a>Ein kurzer \u00dcberblick<\/h4> <p class=\"noindent\">Wir fassen die behandelten Begriffe nochmals in einem Diagram zusammen. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=848&amp;url=Pictures\/metrik\/ueberblick.pdf\"><img id=\"z71756c62da7c\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/metrik\/ueberblick.svg\" width=\"848\"><\/div>  <\/div> <p class=\"indent\">Wir werden uns im zweiten Semester vor allem mit <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> (mehrdimensionale Analysis) besch\u00e4ftigen; allerdings werden wir auch <math display=\"inline\"><mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <\/math> wie in Beispiel <a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-138002r7\">5.7<\/a> oder gewisse Teilmengen von <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> verwenden. Insbesondere bieten metrische R\u00e4ume den f\u00fcr uns geeigneten allgemeinen Rahmen. <a id=\"x1-141001r141\"><\/a> <\/p> <h4 id=\"z3a12ada7267b\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.3 <\/span> <a id=\"x1-1420003\"><\/a>Offene B\u00e4lle<\/h4> <p class=\"noindent\">Mit dem Abstandsbegriff gegeben durch Metriken lassen sich in Analogie zu Definition&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-61006r52\">2.52<\/a> B\u00e4lle definieren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd42cca70225b\"> <a id=\"x1-142001r16\"><\/a> <span class=\"ecbx-1095\">Definition 5.16 <\/span>(Offene B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> ein metrischer Raum. F\u00fcr ein <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und einen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> nennt man                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">den <span class=\"ecbx-1095\">offenen Ball <\/span>mit Radius <math display=\"inline\"><mi>r<\/mi><\/math> um <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass eine Teilmenge <math display=\"inline\"><mi>O<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecbx-1095\">offen<\/span> ist, falls es zu jedem <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>O<\/mi><\/math> ein <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>O<\/mi><\/math> gibt. <\/p> <\/div> <p class=\"indent\">Wir zeigen im Folgenden, dass der Durchschnitt zweier offener B\u00e4lle offen ist, aber verschieben eine ausf\u00fchrlichere Diskussion dieses und verwandter Begriffe auf das zweite Semester. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z26a003a351de\"> <a id=\"x1-142002r17\"><\/a> <span class=\"ecbx-1095\">Lemma 5.17 <\/span>(Schnitte offener B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer<\/span> <span class=\"ecti-1095\">Raum, seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">offen, das heisst, es<\/span> <span class=\"ecti-1095\">existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir setzen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>r<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und bemerken, dass <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ist, da <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> nach Annahme an <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Es bleibt zu zeigen, dass <math display=\"inline\"><mi>r<\/mi><\/math> die gew\u00fcnschte Eigenschaft erf\u00fcllt. Sei also <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt nach der Dreiecksungleichung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und genauso <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> was das Lemma beweist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-142003r142\"><\/a> <h4 id=\"z4230606f8efe\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.4 <\/span> <a id=\"x1-1430004\"><\/a>Wie sehen metrische R\u00e4ume aus?<\/h4> <p class=\"noindent\">Da metrische R\u00e4ume sozusagen geometrische Objekte darstellen, dr\u00e4ngt sich vielleicht die Frage im Titel des Unterabschnittes auf. Doch ist diese Frage genauso wenig sinnvoll wie zum Beispiel die Frage \u201eWelche Eigenschaften haben chemische Elemente?\u201c. In beiden F\u00e4llen h\u00e4ngt die Antwort stark vom betrachteten Spezialfall ab. Zum Beispiel inkludiert die Frage \u201e Wie sehen metrische R\u00e4ume aus?\u201c auch die Frage \u201eWie sehen Teilmengen von <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> aus?\u201c, denn jede Teilmenge von <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> kann als eigenst\u00e4ndiger metrischer Raum (mit der induzierten Metrik) betrachtet werden. Insbesondere hat auf Grund ihrer Allgemeinenheit diese Frage kaum eine vern\u00fcnftige Antwort. <\/p><p class=\"indent\">Damit Sie sich von der Vielfalt der Antwortm\u00f6glichkeiten ein besseres Bild machen k\u00f6nnen, betrachten wir im Folgenden offene B\u00e4lle in einigen wenigen metrischen R\u00e4umen. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z83e86b86b662\"> <a id=\"x1-143001r18\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.18 <\/span>(Offene B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beschreiben Sie die offenen B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle in folgenden metrischen R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und die diskrete Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit der Manhattanmetrik.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit der franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Eisenbahnmetrik.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit der Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben durch <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit der Metrik induziert durch die Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <p class=\"indent\"><\/p><h4 id=\"zd1e16eaa84d3\"> <a id=\"x1-143002r19\"><\/a> <span class=\"ecbx-1095\">Applet 5.19 <\/span>(B\u00e4lle in einigen metrischen R\u00e4umen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die folgenden Apps sollten helfen, die Vielfalt der M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Gestalt von<\/span> <span class=\"ecti-1095\">B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llen in metrischen R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen zu visualisieren.  <\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"560px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/GZr2KhNW\/width\/688\/height\/560\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"560px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/ZFqYeFR4\/width\/688\/height\/560\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"572px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/zYvnZbSE\/width\/688\/height\/572\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"556px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/rxEUJD97\/width\/688\/height\/556\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <a id=\"x1-143003r139\"><\/a> <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: 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border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 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font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z246d82a5a0f9\" class=\"sectionHead\"><span class=\"titlemark\">5.2 <\/span> <a id=\"x1-1390002\"><\/a>Metrische R\u00e4ume<\/h3> <a id=\"x1-139001r138\"><\/a> <h4 id=\"z1ba3e55997a7\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.1 <\/span> <a id=\"x1-1400001\"><\/a>Definition und erste Beispiele<\/h4> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z46668cc6c766\"> <a id=\"x1-140001r10\"><\/a> <span class=\"ecbx-1095\">Definition 5.10 <\/span>(Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Ein <span class=\"ecbx-1095\">metrischer Raum <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> ist eine Menge <math display=\"inline\"><mi>X<\/mi><\/math> gemeinsam mit einer Abbildung <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> die die <span class=\"ecbx-1095\">Metrik <\/span>auf <math display=\"inline\"><mi>X<\/mi><\/math> genannt wird und die folgenden drei Eigenschaften erf\u00fcllt: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Definitheit) F\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Symmetrie) F\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Dreiecksungleichung) F\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <p class=\"indent\">Intuitiv ausgedr\u00fcckt weist eine Metrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> auf einer Menge <math display=\"inline\"><mi>X<\/mi><\/math> je zwei Punkten ihre <span class=\"ecbx-1095\">Distanz <\/span>(ihren <span class=\"ecbx-1095\">Abstand<\/span>) zu. In dieser Auffassung besagt die Definitheit der Metrik, dass der einzige Punkt, der Abstand Null zu einem gegebenen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> hat, <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> selbst ist. Symmetrie der Metrik besagt, dass der Abstand von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> zu <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> der gleiche ist wie von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> zu <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Fasst man die Distanz zwischen zwei Punkten als die L\u00e4nge eines k\u00fcrzesten Weges vom einen zum anderen Punkt auf (was nicht immer m\u00f6glich ist), dann                                                                                                                                                                           besagt die Dreiecksungleichung, dass die L\u00e4nge eines k\u00fcrzesten Weges von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> nach <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/math> h\u00f6chstens so gross ist wie die L\u00e4nge eines Weges, den man abl\u00e4uft, wenn man zuerst den Umweg nach <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> und von dort aus nach <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <\/math> geht. <\/p><p class=\"indent\">Folgende Beispiele von Metriken sind uns eigentlich bereits bekannt \u2013 siehe Lemma <a href=\"..\/..\/chapter\/metrische-raeume#x1-140002r11\">5.11<\/a> unten: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math> mit der Standardmetrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> definiert durch <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math> mit der Standardmetrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> definiert durch <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> mit der Einsmetrik <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> definiert durch <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn> <\/mrow>  <mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> mit der euklidischen Metrik <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> definiert durch <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn> <\/mrow>  <mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> mit der Maximumsmetrik <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> definiert durch <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn> <\/mrow>  <mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> und damit auch <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> werden wir im Normalfall die euklidische Metrik <math display=\"inline\"><msub><mrow><mi>d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> ben\u00fctzen und diese auch einfach mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> bezeichnen.                                                                                                                                                                           <\/p><p class=\"indent\">Wie wir auch sehen werden, gibt es viele weitere, interessante Beispiele von metrischen R\u00e4umen. Manche aber nicht alle dieser erhalten wir mittels Normen <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><\/math> auf Vektorr\u00e4umen wie in Definition&nbsp;<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-136001r1\">5.1<\/a>. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2787e27b433e\"> <a id=\"x1-140002r11\"><\/a> <span class=\"ecbx-1095\">Lemma 5.11 <\/span>(Eine Norm definiert eine Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">ein Vektorraum <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Hier und auch im Folgenden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir ebenso Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">betrachten,<\/span> <span class=\"ecti-1095\">doch inkludiert der Fall der reellen Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume auch den Fall von komplexen Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen, weshalb wir<\/span> <span class=\"ecti-1095\">Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">hier und im Folgenden nicht mehr getrennt erw<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnen werden.<\/span><\/span><\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><\/math> <span class=\"ecti-1095\">eine Norm<\/span> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann definiert<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Metrik <\/span><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>V<\/mi> <\/math><span class=\"ecti-1095\">, die man auch die<\/span> <span class=\"ecti-1095\">von der Norm <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> <span class=\"ecbi-1095\">induzierte Metrik <\/span><span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">nennt.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Es gilt f\u00fcr <math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach Definitheit der Norm <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Homogenit\u00e4t der Norm f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> gilt f\u00fcr <math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und somit erhalten wir die Symmetrie von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Zuletzt verwenden wir die Dreiecksungleichung der Norm und erhalten                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> Dies zeigt die Dreiecksungleichung f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> womit also <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/msub><\/math> eine Metrik auf <math display=\"inline\"><mi>V<\/mi> <\/math> ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Nicht jede Metrik auf einem Vektorraum muss durch eine Norm gegeben sein. Des Weiteren ist das Messen von Distanzen nicht nur auf Vektorr\u00e4umen von Interesse. Interessante Beispiele dieser Art m\u00f6chten wir nun besprechen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9d35aa01af6c\"> <a id=\"x1-140003r12\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.12 <\/span>(Weitere metrische R\u00e4ume)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Diskrete Metriken) Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge und <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer<\/span> <span class=\"ecti-1095\">Raum. In der Tat ist <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definit und symmetrisch per Definition. Des Weiteren erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">die Dreiecksungleichung:<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">Punkte<\/span> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mi>X<\/mi><\/math><span class=\"ecti-1095\">. Falls<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gilt, dann ist<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">trivialerweise<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt. Falls <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist mindestens von<\/span> <span class=\"ecti-1095\">einem Punkt in <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">verschieden und die Dreiecksungleichung gilt ebenso.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Man beachte, dass die diskrete Metrik auf<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">nicht durch eine Norm gegeben ist. In der Tat w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde eine Norm<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">widerspr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chlicherweise die Homogenit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tseigenschaft in Definition <\/span><a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-136001r1\"><span class=\"ecti-1095\">5.1<\/span><\/a> <span class=\"ecti-1095\">nicht erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Manhattanmetrik) Wir setzen <\/span><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo> <\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">. In der Tat<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">alle Axiome<\/span> <span class=\"ecti-1095\">einer Metrik auf <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">, da<\/span> <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>X<\/mi><mo class=\"MathClass-bin\">\u00d7<\/mo><mi>X<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">die Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung<\/span> <span class=\"ecti-1095\">der Einsmetrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist. Die<\/span> <span class=\"ecti-1095\">Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">wird oft auch Manhattan-Metrik genannt. Grund daf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ist, dass man in<\/span> <span class=\"ecti-1095\">schachbrettartig angelegten Orten wie zum Beispiel Manhattan auf folgende Weise von<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gelangt: Man geht zuerst bei<\/span> <span class=\"ecti-1095\">gleichbleibender <\/span><math display=\"inline\"><mi>y<\/mi><\/math><span class=\"ecti-1095\">-Koordinate<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und dann bei<\/span> <span class=\"ecti-1095\">gleichbleibender <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Koordinate<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">, oder<\/span> <span class=\"ecti-1095\">umgekehrt von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">dann von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Es g<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">be zwar noch andere M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten, aber wenn alle Strassen<\/span> <span class=\"ecti-1095\">in Manhattan von West-Ost oder Nord-S<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">d verlaufen, dann misst<\/span> <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">NY<\/mi><mo>  <\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">den relevanten Abstand zwischen zwei Punkten.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Metrik der franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Eisenbahn) Wir setzen<\/span> <math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und definieren<\/span> <span class=\"ecti-1095\">die SNCF-Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">SNCF<\/mi><mo>  <\/mo> <\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">SNCF<\/mi><mo>  <\/mo> <\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;linear&nbsp;abh\u00e4ngig&nbsp;\u00fcber&nbsp;<\/mtext><\/mstyle><mi>\u211d<\/mi><mstyle class=\"text\"><mtext>&nbsp;sind<\/mtext><\/mstyle> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;linear&nbsp;unabh\u00e4ngig&nbsp;\u00fcber&nbsp;<\/mtext><\/mstyle><mi>\u211d<\/mi><mstyle class=\"text\"><mtext>&nbsp;sind<\/mtext><\/mstyle><\/mtd><\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">. Der<\/span> <span class=\"ecti-1095\">Grund f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Namen dieser Metrik (siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140008r13\"><span class=\"ecti-1095\">5.13<\/span><\/a><span class=\"ecti-1095\">) ist, dass eine Bahnreise von einer franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Stadt bei<\/span> <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">zu einer anderen bei<\/span> <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">meist <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber den Ursprung<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">(auch Paris genannt)<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt, ausser wenn <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">auf derselben \u2013 von Paris ausgehenden geraden Strecke liegen. Gewissermassen besteht<\/span> <math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">in dieser Metrik also aus unendlich vielen Halbgeraden, die sich nur im Ursprung<\/span> <span class=\"ecti-1095\">treffen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Ein kombinatorischer Graph ist eine endliche Menge von Punkten, die sogenannten Ecken, von<\/span> <span class=\"ecti-1095\">welchen einige mit sogenannten Kanten verbunden sind. Diese lassen sich auf nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche<\/span> <span class=\"ecti-1095\">Weise mit mehreren Metriken ausstatten; der Konkretheit halber betrachten wir<\/span> <span class=\"ecti-1095\">einen spezifischen Graphen, doch muss ein Graph nicht unbedingt als Teilmenge<\/span> <span class=\"ecti-1095\">von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">gegeben sein.<\/span> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=86&amp;url=Pictures\/metrik\/graph.pdf\"><img decoding=\"async\" id=\"z6f5e09d592f2\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/metrik\/graph.svg\" width=\"86\" \/><\/div>  <\/div> <p class=\"noindent\"><span class=\"ecti-1095\">Man kann nun eine Metrik auf den Ecken (durch<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u2219<\/mo><\/math> <span class=\"ecti-1095\">gekennzeichnet) dadurch definieren, dass man benachbarten Ecken die Distanz<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">zuweist und dies iteriert. Beispielsweise definiert man die Distanz zweier<\/span> <span class=\"ecti-1095\">Ecken, die man <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber zwei aber nicht weniger Kanten erreichen kann,<\/span> <span class=\"ecti-1095\">als<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dazu notwendig ist, dass man von einer Ecke zu jeder anderen Ecke <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber Ablaufen von<\/span> <span class=\"ecti-1095\">Kanten gelangen kann (wie bei obigem Graphen) \u2013 diese Eigenschaft nennt sich auch<\/span> <span class=\"ecti-1095\">Zusammenhang des Graphen.<\/span> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren ist es auch m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glich, eine Metrik auf dem kompletten<\/span> <span class=\"ecti-1095\">(kontinuierlichen) Graphen zu definieren, indem man die obige Definition auf<\/span> <span class=\"ecti-1095\">folgende Weise erweitert. Fasst man eine Kante als Kopie des Intervalles<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">auf,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">die zwei Ecken der Kante sind, so kann man eine Distanz auf den Kanten <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Distanz<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">definieren. <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">hnlich wie oben kann man nun damit eine Metrik auf dem gesamten Graphen<\/span> <span class=\"ecti-1095\">(inklusive den Kanten) definieren.<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zba227d1040eb\"> <a id=\"x1-140008r13\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.13.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die in Beispiel <\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140003r12\"><span class=\"ecti-1095\">5.12<\/span><\/a><span class=\"ecti-1095\">(iii) definierten Metriken tats<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chlich Metriken sind.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren Sie des Weiteren die Konstruktion der Metriken in (iv) vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndig und formal durch.<\/span> <\/p> <\/div> <p class=\"indent\">Wir bemerken, dass f\u00fcr eine gegebene Teilmenge <math display=\"inline\"><mi>Y<\/mi> <\/math> eines metrischen Raumes <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> die Einschr\u00e4nkung <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>Y<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo><mi>Y<\/mi> <\/mrow> <\/msub> <\/math> eine Metrik auf <math display=\"inline\"><mi>Y<\/mi> <\/math> definiert. Wenn <math display=\"inline\"><mi>Y<\/mi> <\/math> mit dieser Metrik versehen ist, nennen wir <math display=\"inline\"><mi>Y<\/mi> <\/math> einen <span class=\"ecbx-1095\">Teilraum<\/span> des metrischen Raumes <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und die Metrik <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>Y<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo><mi>Y<\/mi> <\/mrow><\/msub><\/math> die <span class=\"ecbx-1095\">induzierte Metrik<\/span>. Wenn nicht anders spezifiziert, statten wir Teilmengen eines metrischen Raumes implizit mit der induzierten Metrik aus. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8b0a4c9e6b0e\"> <a id=\"x1-140009r14\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.14 <\/span>(Umgekehrte Dreiecksungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer Raum.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7b232840b8d1\"> <a id=\"x1-140010r15\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.15 <\/span>(Deformation der Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer Raum. Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2215<\/mo><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt><mstyle class=\"mbox\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mover accent=\"true\"><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Metriken <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>d<\/mi><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mover accent=\"true\"><mrow><mi>d<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> <span class=\"ecti-1095\">auf<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">definieren.<\/span> <\/p> <\/div> <a id=\"x1-140011r140\"><\/a> <h4 id=\"zfab30b961798\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.2 <\/span> <a id=\"x1-1410002\"><\/a>Ein kurzer \u00dcberblick<\/h4> <p class=\"noindent\">Wir fassen die behandelten Begriffe nochmals in einem Diagram zusammen. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=848&amp;url=Pictures\/metrik\/ueberblick.pdf\"><img decoding=\"async\" id=\"z71756c62da7c\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/metrik\/ueberblick.svg\" width=\"848\" \/><\/div>  <\/div> <p class=\"indent\">Wir werden uns im zweiten Semester vor allem mit <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> (mehrdimensionale Analysis) besch\u00e4ftigen; allerdings werden wir auch <math display=\"inline\"><mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <\/math> wie in Beispiel <a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-138002r7\">5.7<\/a> oder gewisse Teilmengen von <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> verwenden. Insbesondere bieten metrische R\u00e4ume den f\u00fcr uns geeigneten allgemeinen Rahmen. <a id=\"x1-141001r141\"><\/a> <\/p> <h4 id=\"z3a12ada7267b\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.3 <\/span> <a id=\"x1-1420003\"><\/a>Offene B\u00e4lle<\/h4> <p class=\"noindent\">Mit dem Abstandsbegriff gegeben durch Metriken lassen sich in Analogie zu Definition&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-61006r52\">2.52<\/a> B\u00e4lle definieren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd42cca70225b\"> <a id=\"x1-142001r16\"><\/a> <span class=\"ecbx-1095\">Definition 5.16 <\/span>(Offene B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/math> ein metrischer Raum. F\u00fcr ein <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und einen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> nennt man                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">den <span class=\"ecbx-1095\">offenen Ball <\/span>mit Radius <math display=\"inline\"><mi>r<\/mi><\/math> um <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass eine Teilmenge <math display=\"inline\"><mi>O<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecbx-1095\">offen<\/span> ist, falls es zu jedem <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>O<\/mi><\/math> ein <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>O<\/mi><\/math> gibt. <\/p> <\/div> <p class=\"indent\">Wir zeigen im Folgenden, dass der Durchschnitt zweier offener B\u00e4lle offen ist, aber verschieben eine ausf\u00fchrlichere Diskussion dieses und verwandter Begriffe auf das zweite Semester. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z26a003a351de\"> <a id=\"x1-142002r17\"><\/a> <span class=\"ecbx-1095\">Lemma 5.17 <\/span>(Schnitte offener B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer<\/span> <span class=\"ecti-1095\">Raum, seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">offen, das heisst, es<\/span> <span class=\"ecti-1095\">existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir setzen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>r<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und bemerken, dass <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ist, da <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> nach Annahme an <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> Es bleibt zu zeigen, dass <math display=\"inline\"><mi>r<\/mi><\/math> die gew\u00fcnschte Eigenschaft erf\u00fcllt. Sei also <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt nach der Dreiecksungleichung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und genauso <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> was das Lemma beweist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-142003r142\"><\/a> <h4 id=\"z4230606f8efe\" class=\"subsectionHead\"><span class=\"titlemark\">5.2.4 <\/span> <a id=\"x1-1430004\"><\/a>Wie sehen metrische R\u00e4ume aus?<\/h4> <p class=\"noindent\">Da metrische R\u00e4ume sozusagen geometrische Objekte darstellen, dr\u00e4ngt sich vielleicht die Frage im Titel des Unterabschnittes auf. Doch ist diese Frage genauso wenig sinnvoll wie zum Beispiel die Frage \u201eWelche Eigenschaften haben chemische Elemente?\u201c. In beiden F\u00e4llen h\u00e4ngt die Antwort stark vom betrachteten Spezialfall ab. Zum Beispiel inkludiert die Frage \u201e Wie sehen metrische R\u00e4ume aus?\u201c auch die Frage \u201eWie sehen Teilmengen von <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> aus?\u201c, denn jede Teilmenge von <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> kann als eigenst\u00e4ndiger metrischer Raum (mit der induzierten Metrik) betrachtet werden. Insbesondere hat auf Grund ihrer Allgemeinenheit diese Frage kaum eine vern\u00fcnftige Antwort. <\/p><p class=\"indent\">Damit Sie sich von der Vielfalt der Antwortm\u00f6glichkeiten ein besseres Bild machen k\u00f6nnen, betrachten wir im Folgenden offene B\u00e4lle in einigen wenigen metrischen R\u00e4umen. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z83e86b86b662\"> <a id=\"x1-143001r18\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.18 <\/span>(Offene B\u00e4lle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beschreiben Sie die offenen B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle in folgenden metrischen R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und die diskrete Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi class=\"qopname\">diskret<\/mi><mo>  <\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit der Manhattanmetrik.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit der franz<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sischen Eisenbahnmetrik.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit der Metrik <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gegeben durch <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit der Metrik induziert durch die Norm <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <\/div> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd1e16eaa84d3\"> <a id=\"x1-143002r19\"><\/a> <span class=\"ecbx-1095\">Applet 5.19 <\/span>(B\u00e4lle in einigen metrischen R\u00e4umen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die folgenden Apps sollten helfen, die Vielfalt der M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Gestalt von<\/span> <span class=\"ecti-1095\">B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llen in metrischen R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen zu visualisieren.  <\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"560px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/GZr2KhNW\/width\/688\/height\/560\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"560px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/ZFqYeFR4\/width\/688\/height\/560\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"572px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/zYvnZbSE\/width\/688\/height\/572\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"556px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/rxEUJD97\/width\/688\/height\/556\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <a id=\"x1-143003r139\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":2,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-63","chapter","type-chapter","status-publish","hentry"],"part":61,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/63\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/63\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=63"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=63"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=63"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}