{"id":62,"date":"2021-12-15T09:53:10","date_gmt":"2021-12-15T09:53:10","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/normierte-vektorraeume\/"},"modified":"2021-12-15T09:53:10","modified_gmt":"2021-12-15T09:53:10","slug":"normierte-vektorraeume","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/normierte-vektorraeume\/","title":{"raw":"Normierte Vektorr\u00e4ume","rendered":"Normierte Vektorr\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: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\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=\"z76d3e11fd99f\" class=\"sectionHead\"><span class=\"titlemark\">5.1 <\/span> <a id=\"x1-1360001\"><\/a>Normierte Vektorr\u00e4ume<\/h3> <p class=\"noindent\">In Kapitel <a href=\"..\/..\/part\/die-reellen-zahlen#x1-430002\">2<\/a> (siehe die Abschnitte <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-600002\">2.4.2<\/a> und <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-610003\">2.4.3<\/a>) haben wir bereits gesehen, wie man Distanzen auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math> messen kann. In Analogie dazu m\u00f6chten wir hier verschiedene Varianten von Normen definieren, welche die Rolle des Absolutbetrags \u00fcbernehmen und Abst\u00e4nde in Vektorr\u00e4umen messen werden. Insbesondere werden wir hier die Vektorr\u00e4ume <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> oder <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr eine im ganzen Abschnitt fixierte Dimension <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> betrachten. Wir schreiben Vektoren in <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> oder <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> in der Form <\/p> <table id=\"zdfb34047ff3e\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi class=\"MathClass-op\">\u22ee<\/mi><mo> <\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">wobei <math display=\"inline\"><mi>t<\/mi><\/math> die \u201e Transposition\u201c des platzsparenden Zeilenvektors zu einem Spaltenvektor bezeichnet. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z2588116221c3\"> <a id=\"x1-136001r1\"><\/a> <span class=\"ecbx-1095\">Definition 5.1 <\/span>(Normen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>V<\/mi> <\/math> ein Vektorraum \u00fcber <math display=\"inline\"><mi>\ud835\udd42<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math> (oder <math display=\"inline\"><mi>\ud835\udd42<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math>). Eine <span class=\"ecbx-1095\">Norm <\/span>auf <math display=\"inline\"><mi>V<\/mi> <\/math> ist eine Abbildung <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2208<\/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 folgende 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\"><mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/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\" \/><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Homogenit\u00e4t) F\u00fcr alle <math display=\"inline\"><mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> und alle <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03b1<\/mi><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/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>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> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><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-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-rel\">\u2225<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <p class=\"noindent\">Man nennt <math display=\"inline\"><mi>V<\/mi> <\/math> gemeinsam mit der Norm <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> auch einen <span class=\"ecbx-1095\">normierten Vektorraum<\/span>. <\/p> <\/div> <p class=\"indent\">Das einfachste Beispiel eines normierten Vektorraum ist wahrscheinlich <math display=\"inline\"><mi>\u211d<\/mi><\/math> (als <math display=\"inline\"><mn>1<\/mn><\/math>-dimensionaler Vektorraum \u00fcber <math display=\"inline\"><mi>\u211d<\/mi><\/math>) mit dem Absolutbetrag <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> (siehe Abschnitt&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-600002\">2.4.2<\/a>). Genauso ist <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit dem Absolutbetrag ein normierter Vektorraum (als Vektorraum \u00fcber <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math>). Folgendes Beispiel ist vielleicht interessanter. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z3c0d68d7dd7e\"> <a id=\"x1-136002r2\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.2 <\/span>(Maximumsnorm und Einsnorm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Zu<\/span> <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">bezeichnen<\/span> <span class=\"ecti-1095\">wir mit <\/span><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">die <\/span><span class=\"ecbi-1095\">Projektion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/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\">auf die <\/span><math display=\"inline\"><mi>j<\/mi><\/math><span class=\"ecti-1095\">-te<\/span> <span class=\"ecti-1095\">Komponente. Die <\/span><span class=\"ecbi-1095\">Maximumsnorm <\/span><span class=\"ecti-1095\">oder <\/span><span class=\"ecbi-1095\">Unendlichnorm<\/span> <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 class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">die <\/span><math display=\"inline\"><mstyle><mn>1<\/mn><\/mstyle><\/math><span class=\"ecti-1095\">-<\/span><span class=\"ecbi-1095\">Norm<\/span> <span class=\"ecti-1095\">ist definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">. Die Maximumsnorm<\/span> <span class=\"ecti-1095\">und die <\/span><math display=\"inline\"><mn>1<\/mn><\/math><span class=\"ecti-1095\">-Norm<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">sind durch die gleichen Formeln definiert (oder <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent dazu durch Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung auf<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">). Wir<\/span> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berlassen Ihnen die <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fung der Eigenschaften in Definition <\/span><a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-136001r1\"><span class=\"ecti-1095\">5.1<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <a id=\"x1-136003r134\"><\/a> <h4 id=\"zdd19d81878b1\" class=\"subsectionHead\"><span class=\"titlemark\">5.1.1 <\/span> <a id=\"x1-1370001\"><\/a>Die euklidsche Norm<\/h4> <p class=\"noindent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir m\u00f6chten nun eine f\u00fcr die sogenannte \u201eEuklidische Geometrie\u201c nat\u00fcrliche Norm auf <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> definieren und besprechen. Das <span class=\"ecbx-1095\">Euklidische innere Produkt <\/span>(oder <span class=\"ecbx-1095\">Skalarprodukt<\/span>) von <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mi>v<\/mi><\/mstyle><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><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\"><mstyle><mi>w<\/mi><\/mstyle><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ist definiert durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mover accent=\"false\" class=\"mml-overline\"><mrow><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dieses erf\u00fcllt folgende Eigenschaften: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Sesquilinearit\u00e4t) F\u00fcr alle <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-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> und <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> gilt <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/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\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Symmetrie) F\u00fcr alle <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> gilt <span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Definitheit) F\u00fcr <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> gilt <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> genau dann, wenn <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist.<\/div><\/div> <p class=\"noindent\">Wir bemerken, dass das Wort \u201esesqui\u201c f\u00fcr eineinhalb steht: das innere Produkt ist linear im ersten Argument und \u201ehalblinear\u201c im zweiten Argument. Das <span class=\"ecbx-1095\">reelle innere Produkt <\/span>auf <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> ist durch dieselbe Formel definiert und erf\u00fcllt an Stelle der Sesquilinearit\u00e4t die <span class=\"ecbx-1095\">Bilinearit<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">t<\/span>, also die Linearit\u00e4t in beiden Argumenten (bei festgehaltenem anderem Argument). <\/p><p class=\"indent\">Den Beweis der Sesquilinearit\u00e4t und der Symmetrie \u00fcberlassen wir als \u00dcbung. Wir beweisen Definitheit. Sei also <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wenn <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist, dann ist auch <span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Wenn <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist, dann muss jeder Summand verschwinden. Also gilt <math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und damit <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Unter Verwendung der obigen Eigenschaften des Euklidschen inneren Produkts l\u00e4sst sich nun eine Norm definieren. Die <span class=\"ecbx-1095\">Euklidsche Norm <\/span>auf <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> ist gegeben durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Sie wird auch die <span class=\"ecbx-1095\">2-Norm <\/span>genannt und dementsprechend als <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><mn>2<\/mn> <\/mrow> <\/msub> <\/math> geschrieben. <\/p><p class=\"indent\">Wir m\u00f6chten im Folgenden zeigen, dass die Euklidsche Norm in der Tat eine Norm ist. Definitheit und Homogenit\u00e4t des Euklidschen Norm folgen direkt aus den Eigenschaften des Euklidschen inneren Produkts (wieso?). Um die Dreiecksungleichung zu beweisen, ben\u00f6tigen wir folgende fundamentale Absch\u00e4tzung. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf6a5b3c6625b\"> <a id=\"x1-137001r3\"><\/a> <span class=\"ecbx-1095\">Proposition 5.3 <\/span>(Cauchy-Schwarz Ungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">. Dann gilt<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <span class=\"ecti-1095\">die Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-137002r1\" \/><mstyle class=\"maketag\"><mtext>(5.1)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren gilt Gleichheit in<\/span>  (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>) <span class=\"ecti-1095\">genau dann, wenn<\/span> <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">linear abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig sind<\/span> <span class=\"ecti-1095\">(das heisst, wenn ein <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">existiert mit <\/span><math display=\"inline\"><mi>\u03b1<\/mi><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <p class=\"indent\">Das innere Produkt zweier Vektoren l\u00e4sst sich also durch die \u201eNormen\u201c der beiden Vektoren auf eine konkrete Art und Weise kontrollieren. Wir merken an, dass der folgende Beweis nur die                                                                                                                                                                           \u201eAxiome\u201c des inneren Produktes Sesquilinearit\u00e4t, Symmetrie und Definitheit und nicht die konkrete Formel in der Definition des Euklidschen inneren Produktes verwendet. <\/p><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\">Falls <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> oder <math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist, so steht auf beiden Seiten von (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>) Null und die Vektoren <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle><\/math> sind linear abh\u00e4ngig. Wir nehmen also an, dass <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gilt f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><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> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><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> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>\u03b1<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>\u03b1<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/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\">Der Ausdruck <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> ist aber nicht-negativ und es folgt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Somit folgt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> was die gew\u00fcnschte Ungleichung (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>) impliziert. Gleichheit gilt genau dann, wenn <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und somit <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/math> ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Alternativ l\u00e4sst sich die Cauchy-Schwarz-Ungleichung auch wie folgt beweisen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5dad7695ab54\"> <a id=\"x1-137003r4\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.4 <\/span>(Cauchy-Ungleichung mit einem <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass alle <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/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\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen und schliessen Sie daraus auf die Cauchy-Schwarz Ungleichung<\/span> (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>)<span class=\"ecti-1095\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Schreiben Sie diese Ungleichung anders und addieren Sie <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi><\/math><span class=\"ecti-1095\">. Nun ersetzen<\/span> <span class=\"ecti-1095\">Sie zuerst <\/span><math display=\"inline\"><mi>v<\/mi><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi><mi>v<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>w<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mi>w<\/mi><\/math><span class=\"ecti-1095\">, und setzen Sie<\/span> <span class=\"ecti-1095\">anschliessend f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>w<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">den<\/span> <span class=\"ecti-1095\">Zahlenwert <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mfrac><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/mfrac><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">ein.<\/span><\/p><\/details>  <\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"za2df10940026\"> <a id=\"x1-137004r5\"><\/a> <span class=\"ecbx-1095\">Korollar 5.5 <\/span>(Euklidische Norm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Euklidische Norm definiert eine Norm auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/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 verbleibt die Dreiecksungleichung zu beweisen. Seien <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> Wir sch\u00e4tzen direkt ab unter Verwendung der Cauchy-Schwarz-Ungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/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\">womit die Aussage nach Ziehen der Wurzel folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Durch Einschr\u00e4nkung auf <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> erhalten wir auch das <span class=\"ecbx-1095\">Euklidische innere Produkt <\/span>und die <span class=\"ecbx-1095\">Euklidische Norm <\/span>auf <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> Alle oben bewiesenen Aussagen gelten analog f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <a id=\"x1-137005r137\"><\/a> <\/p> <h4 id=\"z30dde06e0a13\" class=\"subsectionHead\"><span class=\"titlemark\">5.1.2 <\/span> <a id=\"x1-1380002\"><\/a>Der Raum der stetigen Funktionen<\/h4> <p class=\"noindent\">Wir kennen bereits einige Normen auf endlich-dimensionalen Vektorr\u00e4umen und werden noch weitere kennenlernen. F\u00fcr die Analysis sind allerdings nicht nur endlich-dimensionale normierte Vektorr\u00e4ume interessant, sondern oft auch unendlich-dimensionale. H\u00e4ufig (zum Beispiel bei der Diskussion von Differentialgleichungen) werden dabei sogenannte Funktionenr\u00e4ume untersucht. <\/p><p class=\"indent\">Als Beispiel hierf\u00fcr betrachten wir in diesem Unterabschnitt ein kompaktes Intervall <math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und den den Vektorraum <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <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> der stetigen reellwertigen Funktionen auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze7abe9434562\"> <a id=\"x1-138001r6\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.6.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                   dass                   der                   Vektorraum<\/span> <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> <span class=\"ecti-1095\">unendlich-dimensional ist.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Eine M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeit ist die folgende. Betrachten Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die stetige Funktion <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi><mi>x<\/mi><\/math> <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-rel\">\u2264<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/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\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass diese linear unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig sind. Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen sie Polynomfunktionen<\/span> <span class=\"ecti-1095\">verwenden. <\/span><\/p><\/details>  <\/div> <p class=\"indent\">In diesem Abschnitt definieren wir zwei verschiedene Normen auf <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> \u2013 die Supremumsnorm und die <math display=\"inline\"><mn>1<\/mn><\/math>-Norm.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z959d65f601c3\"> <a id=\"x1-138002r7\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.7 <\/span>(Supremumsnorm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir definieren f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/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\"><span class=\"ecti-1095\">unter Verwendung von Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-100001r69\"><span class=\"ecti-1095\">3.69<\/span><\/a><span class=\"ecti-1095\">. Wir behaupten nun, dass<\/span> <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 class=\"ecti-1095\">eine Norm auf<\/span> <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> <span class=\"ecti-1095\">ist. Es gilt Definitheit,<\/span> <span class=\"ecti-1095\">denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">per<\/span> <span class=\"ecti-1095\">Definition von <\/span><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 class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau<\/span> <span class=\"ecti-1095\">dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <span class=\"ecti-1095\">gilt des Weiteren<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03b1<\/mi><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/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\">und somit Homogenit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/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> <span class=\"ecti-1095\">Schlussendlich gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <span class=\"ecti-1095\">auch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><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><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/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\"><span class=\"ecti-1095\">womit wir die Dreiecksungleichung bewiesen haben und gezeigt haben, dass<\/span> <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 class=\"ecti-1095\">eine Norm<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zde0b271fc007\"> <a id=\"x1-138003r8\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.8 <\/span>(<math display=\"inline\"><mn>1<\/mn><\/math>-Norm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Setze f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/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\">Hier verwenden wir, dass <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/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-rel\">\u21a6<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">stetig ist nach Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94011r52\"><span class=\"ecti-1095\">3.52<\/span><\/a> <span class=\"ecti-1095\">und dass stetige Funktionen Riemann-integrierbar sind (Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123001r42\"><span class=\"ecti-1095\">4.42<\/span><\/a><span class=\"ecti-1095\">). Dann<\/span> <span class=\"ecti-1095\">ist <\/span><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><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Norm auf <\/span><span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z6f45390620f2\"> <a id=\"x1-138004r9\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.9.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen                                           Sie,                                           dass<\/span> <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><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">in                  der                  Tat                  eine                  Norm                  auf<\/span> <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> <span class=\"ecti-1095\">ist.<\/span> <\/p> <\/div> <a id=\"x1-138005r136\"><\/a> \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: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; 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=\"z76d3e11fd99f\" class=\"sectionHead\"><span class=\"titlemark\">5.1 <\/span> <a id=\"x1-1360001\"><\/a>Normierte Vektorr\u00e4ume<\/h3> <p class=\"noindent\">In Kapitel <a href=\"..\/..\/part\/die-reellen-zahlen#x1-430002\">2<\/a> (siehe die Abschnitte <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-600002\">2.4.2<\/a> und <a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-610003\">2.4.3<\/a>) haben wir bereits gesehen, wie man Distanzen auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math> messen kann. In Analogie dazu m\u00f6chten wir hier verschiedene Varianten von Normen definieren, welche die Rolle des Absolutbetrags \u00fcbernehmen und Abst\u00e4nde in Vektorr\u00e4umen messen werden. Insbesondere werden wir hier die Vektorr\u00e4ume <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> oder <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr eine im ganzen Abschnitt fixierte Dimension <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> betrachten. Wir schreiben Vektoren in <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> oder <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> in der Form <\/p> <table id=\"zdfb34047ff3e\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi class=\"MathClass-op\">\u22ee<\/mi><mo> <\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">wobei <math display=\"inline\"><mi>t<\/mi><\/math> die \u201e Transposition\u201c des platzsparenden Zeilenvektors zu einem Spaltenvektor bezeichnet. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2588116221c3\"> <a id=\"x1-136001r1\"><\/a> <span class=\"ecbx-1095\">Definition 5.1 <\/span>(Normen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>V<\/mi> <\/math> ein Vektorraum \u00fcber <math display=\"inline\"><mi>\ud835\udd42<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math> (oder <math display=\"inline\"><mi>\ud835\udd42<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi><\/math>). Eine <span class=\"ecbx-1095\">Norm <\/span>auf <math display=\"inline\"><mi>V<\/mi> <\/math> ist eine Abbildung <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2208<\/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 folgende 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\"><mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/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\" \/><mi>v<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Homogenit\u00e4t) F\u00fcr alle <math display=\"inline\"><mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> und alle <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03b1<\/mi><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi><mo class=\"MathClass-rel\">\u2225<\/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>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> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2225<\/mo><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-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-rel\">\u2225<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/div><\/div> <p class=\"noindent\">Man nennt <math display=\"inline\"><mi>V<\/mi> <\/math> gemeinsam mit der Norm <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> auch einen <span class=\"ecbx-1095\">normierten Vektorraum<\/span>. <\/p> <\/div> <p class=\"indent\">Das einfachste Beispiel eines normierten Vektorraum ist wahrscheinlich <math display=\"inline\"><mi>\u211d<\/mi><\/math> (als <math display=\"inline\"><mn>1<\/mn><\/math>-dimensionaler Vektorraum \u00fcber <math display=\"inline\"><mi>\u211d<\/mi><\/math>) mit dem Absolutbetrag <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> (siehe Abschnitt&nbsp;<a href=\"..\/..\/chapter\/intervalle-und-der-absolutbetrag#x1-600002\">2.4.2<\/a>). Genauso ist <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit dem Absolutbetrag ein normierter Vektorraum (als Vektorraum \u00fcber <math display=\"inline\"><mi>\u211d<\/mi><\/math> oder <math display=\"inline\"><mi>\u2102<\/mi><\/math>). Folgendes Beispiel ist vielleicht interessanter. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3c0d68d7dd7e\"> <a id=\"x1-136002r2\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.2 <\/span>(Maximumsnorm und Einsnorm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Zu<\/span> <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">bezeichnen<\/span> <span class=\"ecti-1095\">wir mit <\/span><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">die <\/span><span class=\"ecbi-1095\">Projektion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/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\">auf die <\/span><math display=\"inline\"><mi>j<\/mi><\/math><span class=\"ecti-1095\">-te<\/span> <span class=\"ecti-1095\">Komponente. Die <\/span><span class=\"ecbi-1095\">Maximumsnorm <\/span><span class=\"ecti-1095\">oder <\/span><span class=\"ecbi-1095\">Unendlichnorm<\/span> <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 class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">die <\/span><math display=\"inline\"><mstyle><mn>1<\/mn><\/mstyle><\/math><span class=\"ecti-1095\">&#8211;<\/span><span class=\"ecbi-1095\">Norm<\/span> <span class=\"ecti-1095\">ist definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo><\/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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">. Die Maximumsnorm<\/span> <span class=\"ecti-1095\">und die <\/span><math display=\"inline\"><mn>1<\/mn><\/math><span class=\"ecti-1095\">-Norm<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">sind durch die gleichen Formeln definiert (oder <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent dazu durch Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung auf<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">). Wir<\/span> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berlassen Ihnen die <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fung der Eigenschaften in Definition <\/span><a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-136001r1\"><span class=\"ecti-1095\">5.1<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <a id=\"x1-136003r134\"><\/a> <h4 id=\"zdd19d81878b1\" class=\"subsectionHead\"><span class=\"titlemark\">5.1.1 <\/span> <a id=\"x1-1370001\"><\/a>Die euklidsche Norm<\/h4> <p class=\"noindent\">Sei <span class=\"maperiod\"><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir m\u00f6chten nun eine f\u00fcr die sogenannte \u201eEuklidische Geometrie\u201c nat\u00fcrliche Norm auf <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> definieren und besprechen. Das <span class=\"ecbx-1095\">Euklidische innere Produkt <\/span>(oder <span class=\"ecbx-1095\">Skalarprodukt<\/span>) von <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mi>v<\/mi><\/mstyle><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><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\"><mstyle><mi>w<\/mi><\/mstyle><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ist definiert durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mover accent=\"false\" class=\"mml-overline\"><mrow><msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dieses erf\u00fcllt folgende Eigenschaften: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Sesquilinearit\u00e4t) F\u00fcr alle <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-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> und <math display=\"inline\"><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> gilt <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/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\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>1<\/mn><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><msub><mrow \/><\/msub><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Symmetrie) F\u00fcr alle <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> gilt <span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">(Definitheit) F\u00fcr <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> gilt <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> genau dann, wenn <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist.<\/div><\/div> <p class=\"noindent\">Wir bemerken, dass das Wort \u201esesqui\u201c f\u00fcr eineinhalb steht: das innere Produkt ist linear im ersten Argument und \u201ehalblinear\u201c im zweiten Argument. Das <span class=\"ecbx-1095\">reelle innere Produkt <\/span>auf <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> ist durch dieselbe Formel definiert und erf\u00fcllt an Stelle der Sesquilinearit\u00e4t die <span class=\"ecbx-1095\">Bilinearit<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">t<\/span>, also die Linearit\u00e4t in beiden Argumenten (bei festgehaltenem anderem Argument). <\/p><p class=\"indent\">Den Beweis der Sesquilinearit\u00e4t und der Symmetrie \u00fcberlassen wir als \u00dcbung. Wir beweisen Definitheit. Sei also <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wenn <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist, dann ist auch <span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Wenn <math display=\"inline\"><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist, dann muss jeder Summand verschwinden. Also gilt <math display=\"inline\"><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und damit <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Unter Verwendung der obigen Eigenschaften des Euklidschen inneren Produkts l\u00e4sst sich nun eine Norm definieren. Die <span class=\"ecbx-1095\">Euklidsche Norm <\/span>auf <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> ist gegeben durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Sie wird auch die <span class=\"ecbx-1095\">2-Norm <\/span>genannt und dementsprechend als <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><mn>2<\/mn> <\/mrow> <\/msub> <\/math> geschrieben. <\/p><p class=\"indent\">Wir m\u00f6chten im Folgenden zeigen, dass die Euklidsche Norm in der Tat eine Norm ist. Definitheit und Homogenit\u00e4t des Euklidschen Norm folgen direkt aus den Eigenschaften des Euklidschen inneren Produkts (wieso?). Um die Dreiecksungleichung zu beweisen, ben\u00f6tigen wir folgende fundamentale Absch\u00e4tzung. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf6a5b3c6625b\"> <a id=\"x1-137001r3\"><\/a> <span class=\"ecbx-1095\">Proposition 5.3 <\/span>(Cauchy-Schwarz Ungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">. Dann gilt<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> <span class=\"ecti-1095\">die Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-137002r1\" \/><mstyle class=\"maketag\"><mtext>(5.1)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren gilt Gleichheit in<\/span>  (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>) <span class=\"ecti-1095\">genau dann, wenn<\/span> <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">linear abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig sind<\/span> <span class=\"ecti-1095\">(das heisst, wenn ein <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">existiert mit <\/span><math display=\"inline\"><mi>\u03b1<\/mi><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <p class=\"indent\">Das innere Produkt zweier Vektoren l\u00e4sst sich also durch die \u201eNormen\u201c der beiden Vektoren auf eine konkrete Art und Weise kontrollieren. Wir merken an, dass der folgende Beweis nur die                                                                                                                                                                           \u201eAxiome\u201c des inneren Produktes Sesquilinearit\u00e4t, Symmetrie und Definitheit und nicht die konkrete Formel in der Definition des Euklidschen inneren Produktes verwendet. <\/p><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\">Falls <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> oder <math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist, so steht auf beiden Seiten von (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>) Null und die Vektoren <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle><\/math> sind linear abh\u00e4ngig. Wir nehmen also an, dass <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gilt f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><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> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><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> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>\u03b1<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u03b1<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>\u03b1<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/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\">Der Ausdruck <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> ist aber nicht-negativ und es folgt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Somit folgt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> was die gew\u00fcnschte Ungleichung (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>) impliziert. Gleichheit gilt genau dann, wenn <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und somit <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><mstyle><mi>w<\/mi><\/mstyle><\/math> ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Alternativ l\u00e4sst sich die Cauchy-Schwarz-Ungleichung auch wie folgt beweisen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5dad7695ab54\"> <a id=\"x1-137003r4\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.4 <\/span>(Cauchy-Ungleichung mit einem <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass alle <\/span><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/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\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llen und schliessen Sie daraus auf die Cauchy-Schwarz Ungleichung<\/span> (<a href=\"..\/..\/chapter\/normierte-vektorraeume#x1-137002r1\">5.1<\/a>)<span class=\"ecti-1095\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Schreiben Sie diese Ungleichung anders und addieren Sie <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi><\/math><span class=\"ecti-1095\">. Nun ersetzen<\/span> <span class=\"ecti-1095\">Sie zuerst <\/span><math display=\"inline\"><mi>v<\/mi><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi><mi>v<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>w<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><msup><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mi>w<\/mi><\/math><span class=\"ecti-1095\">, und setzen Sie<\/span> <span class=\"ecti-1095\">anschliessend f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>w<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">den<\/span> <span class=\"ecti-1095\">Zahlenwert <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mfrac><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><\/mfrac><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">ein.<\/span><\/p><\/details>  <\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za2df10940026\"> <a id=\"x1-137004r5\"><\/a> <span class=\"ecbx-1095\">Korollar 5.5 <\/span>(Euklidische Norm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Euklidische Norm definiert eine Norm auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/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 verbleibt die Dreiecksungleichung zu beweisen. Seien <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo> <mstyle> <mi>w<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> Wir sch\u00e4tzen direkt ab unter Verwendung der Cauchy-Schwarz-Ungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>v<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">|<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>w<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>v<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>w<\/mi><\/mstyle><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/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\">womit die Aussage nach Ziehen der Wurzel folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Durch Einschr\u00e4nkung auf <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> erhalten wir auch das <span class=\"ecbx-1095\">Euklidische innere Produkt <\/span>und die <span class=\"ecbx-1095\">Euklidische Norm <\/span>auf <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> Alle oben bewiesenen Aussagen gelten analog f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <a id=\"x1-137005r137\"><\/a> <\/p> <h4 id=\"z30dde06e0a13\" class=\"subsectionHead\"><span class=\"titlemark\">5.1.2 <\/span> <a id=\"x1-1380002\"><\/a>Der Raum der stetigen Funktionen<\/h4> <p class=\"noindent\">Wir kennen bereits einige Normen auf endlich-dimensionalen Vektorr\u00e4umen und werden noch weitere kennenlernen. F\u00fcr die Analysis sind allerdings nicht nur endlich-dimensionale normierte Vektorr\u00e4ume interessant, sondern oft auch unendlich-dimensionale. H\u00e4ufig (zum Beispiel bei der Diskussion von Differentialgleichungen) werden dabei sogenannte Funktionenr\u00e4ume untersucht. <\/p><p class=\"indent\">Als Beispiel hierf\u00fcr betrachten wir in diesem Unterabschnitt ein kompaktes Intervall <math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und den den Vektorraum <math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <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> der stetigen reellwertigen Funktionen auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze7abe9434562\"> <a id=\"x1-138001r6\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.6.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                   dass                   der                   Vektorraum<\/span> <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> <span class=\"ecti-1095\">unendlich-dimensional ist.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Eine M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeit ist die folgende. Betrachten Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die stetige Funktion <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi><mi>x<\/mi><\/math> <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-rel\">\u2264<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/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\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass diese linear unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig sind. Alternativ k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen sie Polynomfunktionen<\/span> <span class=\"ecti-1095\">verwenden. <\/span><\/p><\/details>  <\/div> <p class=\"indent\">In diesem Abschnitt definieren wir zwei verschiedene Normen auf <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> \u2013 die Supremumsnorm und die <math display=\"inline\"><mn>1<\/mn><\/math>-Norm.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z959d65f601c3\"> <a id=\"x1-138002r7\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.7 <\/span>(Supremumsnorm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir definieren f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/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\"><span class=\"ecti-1095\">unter Verwendung von Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-100001r69\"><span class=\"ecti-1095\">3.69<\/span><\/a><span class=\"ecti-1095\">. Wir behaupten nun, dass<\/span> <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 class=\"ecti-1095\">eine Norm auf<\/span> <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> <span class=\"ecti-1095\">ist. Es gilt Definitheit,<\/span> <span class=\"ecti-1095\">denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">per<\/span> <span class=\"ecti-1095\">Definition von <\/span><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 class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau<\/span> <span class=\"ecti-1095\">dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <span class=\"ecti-1095\">gilt des Weiteren<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03b1<\/mi><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/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\">und somit Homogenit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/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> <span class=\"ecti-1095\">Schlussendlich gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <span class=\"ecti-1095\">auch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><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><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/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\"><span class=\"ecti-1095\">womit wir die Dreiecksungleichung bewiesen haben und gezeigt haben, dass<\/span> <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 class=\"ecti-1095\">eine Norm<\/span> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>V<\/mi> <\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zde0b271fc007\"> <a id=\"x1-138003r8\"><\/a> <span class=\"ecbx-1095\">Beispiel 5.8 <\/span>(<math display=\"inline\"><mn>1<\/mn><\/math>-Norm)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Setze f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/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\">Hier verwenden wir, dass <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/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-rel\">\u21a6<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">stetig ist nach Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94011r52\"><span class=\"ecti-1095\">3.52<\/span><\/a> <span class=\"ecti-1095\">und dass stetige Funktionen Riemann-integrierbar sind (Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123001r42\"><span class=\"ecti-1095\">4.42<\/span><\/a><span class=\"ecti-1095\">). Dann<\/span> <span class=\"ecti-1095\">ist <\/span><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><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Norm auf <\/span><span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6f45390620f2\"> <a id=\"x1-138004r9\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.9.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen                                           Sie,                                           dass<\/span> <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><mn>1<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">in                  der                  Tat                  eine                  Norm                  auf<\/span> <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> <span class=\"ecti-1095\">ist.<\/span> <\/p> <\/div> <a id=\"x1-138005r136\"><\/a> \n","protected":false},"author":1089,"menu_order":1,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-62","chapter","type-chapter","status-publish","hentry"],"part":61,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/62","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\/62\/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\/62\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=62"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=62"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=62"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}