{"id":66,"date":"2021-12-15T09:53:12","date_gmt":"2021-12-15T09:53:12","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-5\/"},"modified":"2021-12-15T09:53:12","modified_gmt":"2021-12-15T09:53:12","slug":"weitere-lernmaterialien-5","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-5\/","title":{"raw":"Weitere Lernmaterialien","rendered":"Weitere Lernmaterialien"},"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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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>\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; 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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=\"z24e18b8a0cc1\" class=\"sectionHead\"><span class=\"titlemark\">5.5 <\/span> <a id=\"x1-1530005\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-153001r152\"><\/a> <h4 id=\"zd157613a4e4f\" class=\"subsectionHead\"><span class=\"titlemark\">5.5.1 <\/span> <a id=\"x1-1540001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Den Konvergenzbegriff, den wir hier eingef\u00fchrt haben, geh\u00f6rt zu den Grundpfeilern der Analysis und wird uns fast st\u00e4ndig begegnen. Weiter werden wir im Folgenden stets von Konvergenz einer Folge 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> sprechen, ohne dabei die Norm zu spezifieren. In Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a> haben wir zum Teil gesehen, wieso dies zul\u00e4ssig ist; auf ein genaueres Argument werden wir im n\u00e4chsten Semester eingehen. <\/p><p class=\"indent\">Zur Definition der Konvergenz ben\u00f6tigten wir den Begriff des Abstands. F\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist dieser nat\u00fcrlich <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <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> verwenden wir meist eine Norm <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> um den Abstand <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> zu definieren. Im Allgemeinen verwenden wir die Metrik <math display=\"inline\"><mi>d<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> auf einem metrischen Raum <math display=\"inline\"><mi>X<\/mi><\/math> um den Abstand <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> zu bestimmen. Anfangs mag diese Unterscheidung verwirrend sein, doch sollte sich die Verwirrung jeweils aufl\u00f6sen wenn man sich an den betrachten Rahmen der Diskussion erinnert. Also zum Beispiel wollen wir nicht <math display=\"inline\"><mi>d<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> schreiben, wenn <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nur reelle Zahlen sind. Umgekehrt macht <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> einfach keinen Sinn wenn <math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><\/math> Ecken in einem kombinatorischen Graphen wie in Beispiel <a href=\"..\/..\/chapter\/metrische-raeume#x1-140003r12\">5.12<\/a>(iv) sind. <\/p><p class=\"indent\">Des Weiteren haben wir den Begriff der Stetigkeit aus Kapitel&nbsp;<a href=\"..\/..\/part\/funktionen-und-die-reellen-zahlen#x1-760003\">3<\/a> auf allgemeine metrische R\u00e4ume verallgemeinert und verschiedene Charakterisierungen angegeben (Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>). Insbesondere steht uns also frei, die Charakterisierung unserer Wahl zu verwenden, wenn wir Stetigkeit einer spezifischen Funktion zeigen wollen. Im n\u00e4chsten Semester werden uns weitere Charakterisierungen (mit Hilfe der sogenannten Topologie) begegnen. <a id=\"x1-154001r154\"><\/a> <\/p> <h4 id=\"zdc3eae2bb2c4\" class=\"subsectionHead\"><span class=\"titlemark\">5.5.2 <\/span> <a id=\"x1-1550002\"><\/a>Weitere \u00dcbungen<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7b36fe3cabde\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Hexagon-Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie eine Norm auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass der Einheitsball <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der induzierten Metrik das regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Hexagon mit Eckpunkt <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist. Gibt es eine Norm auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass der Einheitsball durch ein regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">res Pentagon gegeben ist?<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd7c284a1a0a9\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Ultrametriken)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine Ultrametrik auf einer Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist eine Abbildung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die die gleichen Eigenschaften wie eine Metrik hat, abgesehen davon, dass sie anstelle der<\/span> <span class=\"ecti-1095\">Dreiecksungleichung die Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/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 alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass jede Ultrametrik eine Metrik ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><math display=\"inline\"><mi>p<\/mi><\/math><span class=\"ecti-1095\">-adische<\/span> <span class=\"ecti-1095\">Metrik auf <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine Primzahl. Wir definieren<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03bd<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle class=\"text\"><mtext>&nbsp;teilt&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><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\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi>\u03bd<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>y<\/mi> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/mtd><\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Ultrametrik auf <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">definiert und beschreiben Sie die B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle in dieser Metrik.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge und<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">eine Ultrametrik. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">der Ball<\/span> <span class=\"ecti-1095\">von Radius <\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">um <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">gleich <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist. In anderen Worten ist jeder Punkt in einem Ball Zentrum dieses Balles.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z040b77788595\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Rangmetrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">ein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper und <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">die Menge der <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>n<\/mi><\/math><span class=\"ecti-1095\">-Matrizen<\/span> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir definieren <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/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\">=<\/mo><mi class=\"qopname\"> rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">eine Metrik auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r              die              Dreiecksungleichung              reicht              es<\/span> <math display=\"inline\"><mi class=\"qopname\">rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                                                                                                         alle<\/span> <math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">zu beweisen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5cb73cdcbbbf\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Beschr\u00e4nktheit konvergenter Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer<\/span> <span class=\"ecti-1095\">Raum und sei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">nennen eine Teilmenge <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt, falls ein <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert mit <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>M<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass obiger Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktheitsbegriff nicht von der Wahl des Punktes <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine konvergente Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung verallgemeinert Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\"><span class=\"ecti-1095\">5.27<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8d0c56f1528b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge<\/span> <span class=\"ecti-1095\">und seien <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Metriken auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen es gibt eine Konstante <\/span><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>C<\/mi><\/mrow><\/mfrac><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>C<\/mi><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-155007r4\" \/><mstyle class=\"maketag\"><mtext>(5.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass eine Folge genau dann bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich<\/span> <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent ist,<\/span> <span class=\"ecti-1095\">wenn sie bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent ist, und dass in diesem Fall die Grenzwerte <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie zwei Metriken auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">deren Konvergenzbegriffe in obigem Sinne <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r welche keine Konstante<\/span> <span class=\"ecti-1095\">wie in<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-155007r4\">5.4<\/a>) <span class=\"ecti-1095\">existiert.<\/span> <p class=\"noindent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140010r15\"><span class=\"ecti-1095\">5.15<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details><\/dd><\/dl> <\/div> <p class=\"indent\"> <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: 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}\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 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=\"z24e18b8a0cc1\" class=\"sectionHead\"><span class=\"titlemark\">5.5 <\/span> <a id=\"x1-1530005\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-153001r152\"><\/a> <h4 id=\"zd157613a4e4f\" class=\"subsectionHead\"><span class=\"titlemark\">5.5.1 <\/span> <a id=\"x1-1540001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Den Konvergenzbegriff, den wir hier eingef\u00fchrt haben, geh\u00f6rt zu den Grundpfeilern der Analysis und wird uns fast st\u00e4ndig begegnen. Weiter werden wir im Folgenden stets von Konvergenz einer Folge 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> sprechen, ohne dabei die Norm zu spezifieren. In Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a> haben wir zum Teil gesehen, wieso dies zul\u00e4ssig ist; auf ein genaueres Argument werden wir im n\u00e4chsten Semester eingehen. <\/p><p class=\"indent\">Zur Definition der Konvergenz ben\u00f6tigten wir den Begriff des Abstands. F\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist dieser nat\u00fcrlich <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <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> verwenden wir meist eine Norm <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> um den Abstand <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">\u2225<\/mo><\/math> zu definieren. Im Allgemeinen verwenden wir die Metrik <math display=\"inline\"><mi>d<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> auf einem metrischen Raum <math display=\"inline\"><mi>X<\/mi><\/math> um den Abstand <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> zu bestimmen. Anfangs mag diese Unterscheidung verwirrend sein, doch sollte sich die Verwirrung jeweils aufl\u00f6sen wenn man sich an den betrachten Rahmen der Diskussion erinnert. Also zum Beispiel wollen wir nicht <math display=\"inline\"><mi>d<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> schreiben, wenn <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> nur reelle Zahlen sind. Umgekehrt macht <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> einfach keinen Sinn wenn <math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><\/math> Ecken in einem kombinatorischen Graphen wie in Beispiel <a href=\"..\/..\/chapter\/metrische-raeume#x1-140003r12\">5.12<\/a>(iv) sind. <\/p><p class=\"indent\">Des Weiteren haben wir den Begriff der Stetigkeit aus Kapitel&nbsp;<a href=\"..\/..\/part\/funktionen-und-die-reellen-zahlen#x1-760003\">3<\/a> auf allgemeine metrische R\u00e4ume verallgemeinert und verschiedene Charakterisierungen angegeben (Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>). Insbesondere steht uns also frei, die Charakterisierung unserer Wahl zu verwenden, wenn wir Stetigkeit einer spezifischen Funktion zeigen wollen. Im n\u00e4chsten Semester werden uns weitere Charakterisierungen (mit Hilfe der sogenannten Topologie) begegnen. <a id=\"x1-154001r154\"><\/a> <\/p> <h4 id=\"zdc3eae2bb2c4\" class=\"subsectionHead\"><span class=\"titlemark\">5.5.2 <\/span> <a id=\"x1-1550002\"><\/a>Weitere \u00dcbungen<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7b36fe3cabde\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Hexagon-Metrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie eine Norm auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass der Einheitsball <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich der induzierten Metrik das regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Hexagon mit Eckpunkt <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist. Gibt es eine Norm auf <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass der Einheitsball durch ein regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">res Pentagon gegeben ist?<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd7c284a1a0a9\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Ultrametriken)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine Ultrametrik auf einer Menge <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist eine Abbildung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die die gleichen Eigenschaften wie eine Metrik hat, abgesehen davon, dass sie anstelle der<\/span> <span class=\"ecti-1095\">Dreiecksungleichung die Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/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 alle <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass jede Ultrametrik eine Metrik ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(<\/span><math display=\"inline\"><mi>p<\/mi><\/math><span class=\"ecti-1095\">-adische<\/span> <span class=\"ecti-1095\">Metrik auf <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">eine Primzahl. Wir definieren<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03bd<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>p<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle class=\"text\"><mtext>&nbsp;teilt&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><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\">\u2208<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u2124<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>p<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi>\u03bd<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>y<\/mi> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/mtd><\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Ultrametrik auf <\/span><math display=\"inline\"><mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">definiert und beschreiben Sie die B<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle in dieser Metrik.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge und<\/span> <math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">eine Ultrametrik. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">der Ball<\/span> <span class=\"ecti-1095\">von Radius <\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">um <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">gleich <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>r<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist. In anderen Worten ist jeder Punkt in einem Ball Zentrum dieses Balles.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z040b77788595\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Rangmetrik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">ein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper und <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">die Menge der <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>n<\/mi><\/math><span class=\"ecti-1095\">-Matrizen<\/span> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir definieren <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/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\">=<\/mo><mi class=\"qopname\"> rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">eine Metrik auf <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r              die              Dreiecksungleichung              reicht              es<\/span> <math display=\"inline\"><mi class=\"qopname\">rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> rang<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                                                                                                         alle<\/span> <math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>B<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">zu beweisen.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5cb73cdcbbbf\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Beschr\u00e4nktheit konvergenter Folgen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer<\/span> <span class=\"ecti-1095\">Raum und sei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">nennen eine Teilmenge <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt, falls ein <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert mit <\/span><math display=\"inline\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>M<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass obiger Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktheitsbegriff nicht von der Wahl des Punktes <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine konvergente Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Diese <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung verallgemeinert Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-145007r27\"><span class=\"ecti-1095\">5.27<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8d0c56f1528b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">eine Menge<\/span> <span class=\"ecti-1095\">und seien <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Metriken auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen es gibt eine Konstante <\/span><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>C<\/mi><\/mrow><\/mfrac><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>C<\/mi><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-155007r4\" \/><mstyle class=\"maketag\"><mtext>(5.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass eine Folge genau dann bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich<\/span> <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent ist,<\/span> <span class=\"ecti-1095\">wenn sie bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent ist, und dass in diesem Fall die Grenzwerte <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie zwei Metriken auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">deren Konvergenzbegriffe in obigem Sinne <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r welche keine Konstante<\/span> <span class=\"ecti-1095\">wie in<\/span> (<a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-155007r4\">5.4<\/a>) <span class=\"ecti-1095\">existiert.<\/span> <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\">Siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140010r15\"><span class=\"ecti-1095\">5.15<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details><\/dd><\/dl> <\/div> <div class=\"wp-nocaption \"><\/div> 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