{"id":50,"date":"2021-12-15T09:53:06","date_gmt":"2021-12-15T09:53:06","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-3\/"},"modified":"2021-12-15T09:53:06","modified_gmt":"2021-12-15T09:53:06","slug":"weitere-lernmaterialien-3","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-3\/","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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\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=\"z29a099bac622\" class=\"sectionHead\"><span class=\"titlemark\">3.9 <\/span> <a id=\"x1-1030009\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-103001r102\"><\/a> <h4 id=\"z0118b6518b81\" class=\"subsectionHead\"><span class=\"titlemark\">3.9.1 <\/span> <a id=\"x1-1040001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Wir werden die Notationen <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und das Verhalten dieser, zum Beispiel unter Indexverschiebung, immer h\u00e4ufiger ben\u00f6tigen. Ebenso sind Polynome, die Fakult\u00e4t, der Binomialsatz und die Begriffe der Monotonie und Stetigkeit f\u00fcr alles Weitere von fundamentaler Bedeutung, weswegen diese Begriffe und die ersten Resultate f\u00fcr diese Begriffe in Zukunft meist ohne Verweis auf die jeweiligen Definitionen oder S\u00e4tze verwendet werden. <\/p><p class=\"indent\">Der Zwischenwertsatz (Satz <a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96001r58\">3.58<\/a>) ist ein wichtiges Resultat. Vor allem aber ist er ein wichtiger Bestandteil unseres Beweises von dem Satz \u00fcber den Umkehrsatz (Satz <a href=\"..\/..\/chapter\/der-satz-ueber-die-umkehrabbildung#x1-97001r64\">3.64<\/a>), welchen wir sp\u00e4ter f\u00fcr die korrekte Konstruktion vieler Funktionen verwenden werden. Insbesondere erlaubt uns letzterer die Funktionen <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr jedes <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msup><\/math> f\u00fcr jedes <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> zu definieren. Wir werden diese und alle dazugeh\u00f6rigen Potenzregeln in \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/der-satz-ueber-die-umkehrabbildung#x1-97003r66\">3.66<\/a> in Zukunft ohne Verweis verwenden. <\/p><p class=\"indent\">Die Resultate aus Abschnitt <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-990008\">3.8<\/a> (also der Satz \u00fcber die Beschr\u00e4nktheit und die gleichm\u00e4ssige Stetigkeit) werden bereits im n\u00e4chsten Kapitel Bedeutung erhalten. Wie wir sp\u00e4ter sehen werden, sind diese Resultate Spezialf\u00e4lle von allgemeineren Aussage f\u00fcr stetige Funktionen auf sogenannten \u201e kompakten metrischen R\u00e4umen\u201c. Mittlerweile sollten Sie logisch geschult sein und den Unterschied (vergleiche Beispiele <a href=\"..\/..\/chapter\/logische-begriffe#x1-8005r6\">1.6<\/a> und <a href=\"..\/..\/chapter\/logische-begriffe#x1-8006r7\">1.7<\/a>) in den Definitionen von Stetigkeit und gleichm\u00e4ssiger Stetigkeit klar erkennen, weswegen Sie auch den Satz \u00fcber die gleichm\u00e4ssige Stetigkeit besonders sch\u00e4tzen sollten. Wir wollen noch betonen, dass diese Unterscheidung keine Spitzfindigkeit darstellt. <a id=\"x1-104001r104\"><\/a> <\/p> <h4 id=\"z59aacf48dada\" class=\"subsectionHead\"><span class=\"titlemark\">3.9.2 <\/span> <a id=\"x1-1050002\"><\/a>Weitere \u00dcbungsaufgaben<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2057a04971dc\"> <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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und seien<\/span> <math display=\"inline\"><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>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">Elemente eines<\/span> <span class=\"ecti-1095\">komplexen Vektorraums <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Finden Sie einen vereinfachten Ausdruck f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Doppelsumme<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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>n<\/mi><\/mrow><\/munderover><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>j<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/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\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd35ef9ceeee6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Formale Definition des Polynomrings)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Das Ziel dieser Aufgabe ist, den Ring der Polynome <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber einem beliebigen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper formal zu definieren. Im<\/span> <span class=\"ecti-1095\">Folgenden ist <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">beliebiger K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper und <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">bezeichnet die Teilmenge <\/span>der schliesslich verschwindenden <span class=\"ecti-1095\">Funktionen in<\/span> <math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\ud835\udd42<\/mi><\/math><span class=\"ecti-1095\">, das<\/span> <span class=\"ecti-1095\">heisst,<\/span> <\/p> <table id=\"z30ff37e44eae\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Des Weiteren definieren wir Operationen <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">\u22c5<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/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\"><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <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>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/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> <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\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/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 <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit den oben definierten Operationen einen kommutativen Ring bildet.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir fassen <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">als eine Teilmenge von <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">auf, indem wir <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">mit der Funktion <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>a<\/mi><msub><mrow><mi>\ud835\udfd9<\/mi><\/mrow> <mrow> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">identifizieren. Zeigen Sie, dass <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">eine Null und <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">eine Eins des Ringes <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><msup><mrow><mi>X<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">die Abbildung gegeben durch <\/span><table id=\"zec2d86dcbe5b\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"><msup><mrow> <mi>X<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>1<\/mn><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>k<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn><mspace class=\"quad\" width=\"1em\" \/><\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>sonst<\/mtext><\/mstyle><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow> <\/math><\/td><\/tr><\/table> <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\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><span class=\"ecti-1095\">. Zeigen Sie,<\/span> <span class=\"ecti-1095\">dass sich jedes Element <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">als eindeutig bestimmten Ausdruck der Form<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>X<\/mi><\/mrow><mrow><mi>n<\/mi><\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und Zahlen <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Vergleichen Sie die obigen Definitionen zur Definition des Polynomrings in Definition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/polynome#x1-81005r13\"><span class=\"ecti-1095\">3.13<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze4f3bfdd0843\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Ein K\u00f6rper mit neun Elementen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit neun Elementen konstruieren und folgen dabei<\/span> <span class=\"ecti-1095\">der Bemerkung am Ende von Abschnitt <\/span><a href=\"..\/..\/chapter\/polynome#x1-830002\"><span class=\"ecti-1095\">3.2.2<\/span><\/a><span class=\"ecti-1095\">.<\/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 das Polynom <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber dem K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">keine Nullstelle besitzt.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Wir betrachten nun den Polynomring <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und die Relation <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u223c<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mstyle class=\"text\"><mtext>&nbsp;teilt&nbsp;<\/mtext><\/mstyle><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ist. Sei <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">der dazugeh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rige Quotientenraum.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Operationen<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/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\">wohldefiniert sind und aus <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit neun Elementen machen.<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2a5e0dd55b35\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Zwei Identit\u00e4ten f\u00fcr Binomialkoeffizienten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow> <mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><\/mfenced><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-bin\">\u2212<\/mo><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow> <mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z49d9bc78a1a4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Nicomachus Theorem)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\"><span class=\"ecti-1095\">3.32<\/span><\/a> <span class=\"ecti-1095\">haben wr die Summe <\/span><math display=\"inline\"><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">als Werte eines<\/span> <span class=\"ecti-1095\">Polynoms vom Grad <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">mit Leitkoeffizient <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">ausgedr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckt. In der Tat existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle Koeffizienten eine Formel \u2013 Faulhaber\u2019s<\/span> <span class=\"ecti-1095\">Formel \u2013 in Termen der sogenannten Bernoulli-Zahlen. Wir verzichten hier auf diese<\/span> <span class=\"ecti-1095\">und beweisen einen Spezialfall \u2013 Nicomachus Theorem. Dieses besagt, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>n<\/mi><\/mrow><\/munderover><mi>k<\/mi><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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\">Beweisen Sie Nicomachus Theorem, indem Sie Abel-Summation wie in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-78001r3\"><span class=\"ecti-1095\">3.3<\/span><\/a> <span class=\"ecti-1095\">anwenden.<\/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\">Betrachten Sie <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <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><mi>k<\/mi><\/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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc9f11e39f62e\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(<math display=\"inline\"><mi>\u211d<\/mi><\/math>-wertige Funktionen auf einer Zweipunktmenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">eine                            Menge                            bestehend                            aus<\/span> <math display=\"inline\"><mn>2<\/mn><\/math> <span class=\"ecti-1095\">Elementen. Zeigen    Sie,    dass    es    einen    Isomorphismus    von    Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen<\/span> <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">gibt. Induzieren      Sie      durch      diese      Bijektion      eine      Ordnung      auf<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und beschreiben  Sie  diese  (beispielsweise  duch  Beschreibung  welche  Elemente  gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sser  als<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und                              welche                              kleiner                              als<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">sind).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z61473a056e48\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Dimension von <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">eine              nicht-leere              Menge.              Zeigen              Sie,              dass<\/span> <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">genau                 dann                 endlich-dimensional                 ist,                 wenn<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">endlich ist      und      dass      in      diesem      Fall      die      Dimension      gerade<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>D<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/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\">Betrachten Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>y<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>sonst<\/mtext><\/mstyle><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und zeigen Sie, dass <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine linear<\/span> <span class=\"ecti-1095\">unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngige Teilmenge von <\/span><math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist. Falls <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>D<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">(und nur dann), bilden diese Funktionen auch eine Basis von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze9561843c4a0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eigenschaften komplexwertiger Funktionen)<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\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine nicht-leere Teilmenge.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Definieren Sie den Begriff der Stetigkeit (in einem Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Funktionen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">genau dann in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">stetig ist, wenn die Funktionen <\/span><math display=\"inline\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">stetig sind.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren Sie das Analogon von Proposition <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94008r50\"><span class=\"ecti-1095\">3.50<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r komplexwertige Funktionen und<\/span> <span class=\"ecti-1095\">beweisen Sie es (zum Beispiel unter Verwendung von (ii) oder direkt).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren und beweisen Sie Proposition <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94011r52\"><span class=\"ecti-1095\">3.52<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r komplexwertige Funktionen.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd9028e36cee8\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Formalisierung der Nicht-Stetigkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion. Dr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken<\/span> <span class=\"ecti-1095\">Sie die Aussagen<\/span> <span class=\"ecti-1095\">\u201e<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist nicht<\/span> <span class=\"ecti-1095\">stetig<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">\u201e<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist nicht<\/span> <span class=\"ecti-1095\">stetig bei einem Punkt <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> <span class=\"ecti-1095\">in Pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">dikatenlogik aus. Zeigen Sie damit, dass die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">aus dem Teilabschnitt <\/span><a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-930002\"><span class=\"ecti-1095\">3.4.2<\/span><\/a> <span class=\"ecti-1095\">nicht stetig ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z913cdd03ef91\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Lineare Absch\u00e4tzung bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion. Angenommen es existiert zu <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">eine Konstante <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>L<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>L<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z03d687b76de9\"> <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>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Teilmenge und seien <\/span><span class=\"maperiod\"><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><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass dann auch die Funktionen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/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\"><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">stetig sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z6b3859d9dfb2\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kompakter Tr\u00e4ger)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir sagen, dass eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">einen kompakten Tr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ger hat, 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>f<\/mi><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 alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion mit kompaktem Tr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ger. Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig stetig und beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za65a658c9138\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Offene und abgeschlossene Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zeigen, dass sich das offene<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">Intervall vom<\/span> <span class=\"ecti-1095\">abgeschlossenen <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">Intervall zwar von der Kardinalit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t her nicht unterscheiden, aber von der Ordnung her sehr<\/span> <span class=\"ecti-1095\">wohl.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie eine Bijektion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass keine stetige, bijektive Abbildung <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">existieren kann.<\/span><\/dd><\/dl> <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\">Entfernen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (ii) einen Punkt aus<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8690f0275de9\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                   dass                   die                   Abbildung<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>7<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bijektiv ist (ohne zu versuchen, eine Formel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die inverse Abbildung anzugeben).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z4c60bfa09c4b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Satz <\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-100001r69\"><span class=\"ecti-1095\">3.69<\/span><\/a> <span class=\"ecti-1095\">und Korollar <\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-101001r71\"><span class=\"ecti-1095\">3.71<\/span><\/a> <span class=\"ecti-1095\">mit Hilfe des Intervallschachtelungsprinzips in<\/span> <span class=\"ecti-1095\">Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-70001r77\"><span class=\"ecti-1095\">2.77<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd82af0946156\"> <a id=\"x1-105014r81\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.81 <\/span>(Challenge: \u201eFast \u00fcberall\u201c Stetigkeit von monotonen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zeigen, dass es zu einer monotonen Funktion<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf einem<\/span> <span class=\"ecti-1095\">Intervall <\/span><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 class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens abz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar viele<\/span> <span class=\"ecti-1095\">Punkte geben kann, bei denen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nicht stetig ist (sogenannte Unstetigkeitsstellen). Gehen Sie dazu wie folgt vor: Sei<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">die Menge der<\/span> <span class=\"ecti-1095\">Unstetigkeitsstellen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/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\">Sei <\/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> <span class=\"ecti-1095\">Wir setzen<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>x<\/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\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">W<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie anschliessend eine rationale Zahl<\/span> <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Wir nehmen hier an, dass wir beliebig oft eine Wahl treffen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.)<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist und schliessen Sie auf die Aussage.<\/span><\/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: 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}\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 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=\"z29a099bac622\" class=\"sectionHead\"><span class=\"titlemark\">3.9 <\/span> <a id=\"x1-1030009\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-103001r102\"><\/a> <h4 id=\"z0118b6518b81\" class=\"subsectionHead\"><span class=\"titlemark\">3.9.1 <\/span> <a id=\"x1-1040001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Wir werden die Notationen <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> und das Verhalten dieser, zum Beispiel unter Indexverschiebung, immer h\u00e4ufiger ben\u00f6tigen. Ebenso sind Polynome, die Fakult\u00e4t, der Binomialsatz und die Begriffe der Monotonie und Stetigkeit f\u00fcr alles Weitere von fundamentaler Bedeutung, weswegen diese Begriffe und die ersten Resultate f\u00fcr diese Begriffe in Zukunft meist ohne Verweis auf die jeweiligen Definitionen oder S\u00e4tze verwendet werden. <\/p><p class=\"indent\">Der Zwischenwertsatz (Satz <a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96001r58\">3.58<\/a>) ist ein wichtiges Resultat. Vor allem aber ist er ein wichtiger Bestandteil unseres Beweises von dem Satz \u00fcber den Umkehrsatz (Satz <a href=\"..\/..\/chapter\/der-satz-ueber-die-umkehrabbildung#x1-97001r64\">3.64<\/a>), welchen wir sp\u00e4ter f\u00fcr die korrekte Konstruktion vieler Funktionen verwenden werden. Insbesondere erlaubt uns letzterer die Funktionen <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup><\/math> f\u00fcr jedes <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msup><\/math> f\u00fcr jedes <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> zu definieren. Wir werden diese und alle dazugeh\u00f6rigen Potenzregeln in \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/der-satz-ueber-die-umkehrabbildung#x1-97003r66\">3.66<\/a> in Zukunft ohne Verweis verwenden. <\/p><p class=\"indent\">Die Resultate aus Abschnitt <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-990008\">3.8<\/a> (also der Satz \u00fcber die Beschr\u00e4nktheit und die gleichm\u00e4ssige Stetigkeit) werden bereits im n\u00e4chsten Kapitel Bedeutung erhalten. Wie wir sp\u00e4ter sehen werden, sind diese Resultate Spezialf\u00e4lle von allgemeineren Aussage f\u00fcr stetige Funktionen auf sogenannten \u201e kompakten metrischen R\u00e4umen\u201c. Mittlerweile sollten Sie logisch geschult sein und den Unterschied (vergleiche Beispiele <a href=\"..\/..\/chapter\/logische-begriffe#x1-8005r6\">1.6<\/a> und <a href=\"..\/..\/chapter\/logische-begriffe#x1-8006r7\">1.7<\/a>) in den Definitionen von Stetigkeit und gleichm\u00e4ssiger Stetigkeit klar erkennen, weswegen Sie auch den Satz \u00fcber die gleichm\u00e4ssige Stetigkeit besonders sch\u00e4tzen sollten. Wir wollen noch betonen, dass diese Unterscheidung keine Spitzfindigkeit darstellt. <a id=\"x1-104001r104\"><\/a> <\/p> <h4 id=\"z59aacf48dada\" class=\"subsectionHead\"><span class=\"titlemark\">3.9.2 <\/span> <a id=\"x1-1050002\"><\/a>Weitere \u00dcbungsaufgaben<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2057a04971dc\"> <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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und seien<\/span> <math display=\"inline\"><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>n<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">Elemente eines<\/span> <span class=\"ecti-1095\">komplexen Vektorraums <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Finden Sie einen vereinfachten Ausdruck f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Doppelsumme<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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>n<\/mi><\/mrow><\/munderover><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>j<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>v<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>v<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/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\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd35ef9ceeee6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Formale Definition des Polynomrings)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Das Ziel dieser Aufgabe ist, den Ring der Polynome <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber einem beliebigen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper formal zu definieren. Im<\/span> <span class=\"ecti-1095\">Folgenden ist <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">beliebiger K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper und <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">bezeichnet die Teilmenge <\/span>der schliesslich verschwindenden <span class=\"ecti-1095\">Funktionen in<\/span> <math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\ud835\udd42<\/mi><\/math><span class=\"ecti-1095\">, das<\/span> <span class=\"ecti-1095\">heisst,<\/span> <\/p> <table id=\"z30ff37e44eae\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Des Weiteren definieren wir Operationen <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">\u22c5<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/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\"><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <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>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><mo class=\"MathClass-close\">)<\/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> <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\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/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 <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit den oben definierten Operationen einen kommutativen Ring bildet.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir fassen <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">als eine Teilmenge von <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">auf, indem wir <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">mit der Funktion <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>a<\/mi><msub><mrow><mi>\ud835\udfd9<\/mi><\/mrow> <mrow> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">identifizieren. Zeigen Sie, dass <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">eine Null und <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">eine Eins des Ringes <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">sei <\/span><math display=\"inline\"><msup><mrow><mi>X<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">die Abbildung gegeben durch <\/span><table id=\"zec2d86dcbe5b\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"><msup><mrow> <mi>X<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>1<\/mn><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>k<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn><mspace class=\"quad\" width=\"1em\" \/><\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>sonst<\/mtext><\/mstyle><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow> <\/math><\/td><\/tr><\/table> <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\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><span class=\"ecti-1095\">. Zeigen Sie,<\/span> <span class=\"ecti-1095\">dass sich jedes Element <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">als eindeutig bestimmten Ausdruck der Form<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>X<\/mi><\/mrow><mrow><mi>n<\/mi><\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und Zahlen <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Vergleichen Sie die obigen Definitionen zur Definition des Polynomrings in Definition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/polynome#x1-81005r13\"><span class=\"ecti-1095\">3.13<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze4f3bfdd0843\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Ein K\u00f6rper mit neun Elementen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit neun Elementen konstruieren und folgen dabei<\/span> <span class=\"ecti-1095\">der Bemerkung am Ende von Abschnitt <\/span><a href=\"..\/..\/chapter\/polynome#x1-830002\"><span class=\"ecti-1095\">3.2.2<\/span><\/a><span class=\"ecti-1095\">.<\/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 das Polynom <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber dem K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">keine Nullstelle besitzt.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Wir betrachten nun den Polynomring <\/span><math display=\"inline\"><msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und die Relation <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u223c<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mstyle class=\"text\"><mtext>&nbsp;teilt&nbsp;<\/mtext><\/mstyle><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"> <mo class=\"MathClass-rel\">\u223c<\/mo><\/math> <span class=\"ecti-1095\">eine <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quivalenzrelation ist. Sei <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\ud835\udd3d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-bin\">\u2215<\/mo><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mo class=\"MathClass-rel\">\u223c<\/mo><\/mstyle><mtext \/><\/mstyle><\/math> <span class=\"ecti-1095\">der dazugeh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rige Quotientenraum.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Operationen<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub><mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mo class=\"MathClass-rel\">\u223c<\/mo><\/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\">wohldefiniert sind und aus <\/span><math display=\"inline\"><mi>\ud835\udd42<\/mi><\/math> <span class=\"ecti-1095\">einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mit neun Elementen machen.<\/span><\/p><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2a5e0dd55b35\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Zwei Identit\u00e4ten f\u00fcr Binomialkoeffizienten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow> <mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><\/mfenced><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-bin\">\u2212<\/mo><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow> <mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z49d9bc78a1a4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Nicomachus Theorem)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\"><span class=\"ecti-1095\">3.32<\/span><\/a> <span class=\"ecti-1095\">haben wr die Summe <\/span><math display=\"inline\"><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">als Werte eines<\/span> <span class=\"ecti-1095\">Polynoms vom Grad <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">mit Leitkoeffizient <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">ausgedr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckt. In der Tat existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle Koeffizienten eine Formel \u2013 Faulhaber\u2019s<\/span> <span class=\"ecti-1095\">Formel \u2013 in Termen der sogenannten Bernoulli-Zahlen. Wir verzichten hier auf diese<\/span> <span class=\"ecti-1095\">und beweisen einen Spezialfall \u2013 Nicomachus Theorem. Dieses besagt, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>n<\/mi><\/mrow><\/munderover><mi>k<\/mi><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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\">Beweisen Sie Nicomachus Theorem, indem Sie Abel-Summation wie in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-78001r3\"><span class=\"ecti-1095\">3.3<\/span><\/a> <span class=\"ecti-1095\">anwenden.<\/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\">Betrachten Sie <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <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><mi>k<\/mi><\/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>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc9f11e39f62e\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(<math display=\"inline\"><mi>\u211d<\/mi><\/math>-wertige Funktionen auf einer Zweipunktmenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">eine                            Menge                            bestehend                            aus<\/span> <math display=\"inline\"><mn>2<\/mn><\/math> <span class=\"ecti-1095\">Elementen. Zeigen    Sie,    dass    es    einen    Isomorphismus    von    Vektorr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">umen<\/span> <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"MathClass-op\">\u2245<\/mi><mo> <\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">gibt. Induzieren      Sie      durch      diese      Bijektion      eine      Ordnung      auf<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und beschreiben  Sie  diese  (beispielsweise  duch  Beschreibung  welche  Elemente  gr<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sser  als<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und                              welche                              kleiner                              als<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">sind).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z61473a056e48\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Dimension von <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">eine              nicht-leere              Menge.              Zeigen              Sie,              dass<\/span> <math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">genau                 dann                 endlich-dimensional                 ist,                 wenn<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">endlich ist      und      dass      in      diesem      Fall      die      Dimension      gerade<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>D<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/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\">Betrachten Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>y<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>sonst<\/mtext><\/mstyle><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und zeigen Sie, dass <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine linear<\/span> <span class=\"ecti-1095\">unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngige Teilmenge von <\/span><math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist. Falls <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>D<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">(und nur dann), bilden diese Funktionen auch eine Basis von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze9561843c4a0\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Eigenschaften komplexwertiger Funktionen)<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\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine nicht-leere Teilmenge.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Definieren Sie den Begriff der Stetigkeit (in einem Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Funktionen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">genau dann in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">stetig ist, wenn die Funktionen <\/span><math display=\"inline\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">stetig sind.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren Sie das Analogon von Proposition <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94008r50\"><span class=\"ecti-1095\">3.50<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r komplexwertige Funktionen und<\/span> <span class=\"ecti-1095\">beweisen Sie es (zum Beispiel unter Verwendung von (ii) oder direkt).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Formulieren und beweisen Sie Proposition <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94011r52\"><span class=\"ecti-1095\">3.52<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r komplexwertige Funktionen.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd9028e36cee8\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Formalisierung der Nicht-Stetigkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion. Dr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">cken<\/span> <span class=\"ecti-1095\">Sie die Aussagen<\/span> <span class=\"ecti-1095\">\u201e<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist nicht<\/span> <span class=\"ecti-1095\">stetig<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">\u201e<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist nicht<\/span> <span class=\"ecti-1095\">stetig bei einem Punkt <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> <span class=\"ecti-1095\">in Pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">dikatenlogik aus. Zeigen Sie damit, dass die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">aus dem Teilabschnitt <\/span><a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-930002\"><span class=\"ecti-1095\">3.4.2<\/span><\/a> <span class=\"ecti-1095\">nicht stetig ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z913cdd03ef91\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Lineare Absch\u00e4tzung bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion. Angenommen es existiert zu <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">eine Konstante <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>L<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>L<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z03d687b76de9\"> <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>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Teilmenge und seien <\/span><span class=\"maperiod\"><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><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass dann auch die Funktionen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">max<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/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\"><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><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-close\">)<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">stetig sind.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6b3859d9dfb2\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kompakter Tr\u00e4ger)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir sagen, dass eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">einen kompakten Tr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ger hat, 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>f<\/mi><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 alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion mit kompaktem Tr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ger. Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig stetig und beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za65a658c9138\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Offene und abgeschlossene Intervalle)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zeigen, dass sich das offene<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">Intervall vom<\/span> <span class=\"ecti-1095\">abgeschlossenen <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">Intervall zwar von der Kardinalit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t her nicht unterscheiden, aber von der Ordnung her sehr<\/span> <span class=\"ecti-1095\">wohl.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie eine Bijektion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass keine stetige, bijektive Abbildung <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">existieren kann.<\/span><\/dd><\/dl> <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\">Entfernen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r (ii) einen Punkt aus<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8690f0275de9\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                   dass                   die                   Abbildung<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>7<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bijektiv ist (ohne zu versuchen, eine Formel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die inverse Abbildung anzugeben).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4c60bfa09c4b\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Satz <\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-100001r69\"><span class=\"ecti-1095\">3.69<\/span><\/a> <span class=\"ecti-1095\">und Korollar <\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-101001r71\"><span class=\"ecti-1095\">3.71<\/span><\/a> <span class=\"ecti-1095\">mit Hilfe des Intervallschachtelungsprinzips in<\/span> <span class=\"ecti-1095\">Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-70001r77\"><span class=\"ecti-1095\">2.77<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd82af0946156\"> <a id=\"x1-105014r81\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.81 <\/span>(Challenge: \u201eFast \u00fcberall\u201c Stetigkeit von monotonen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zeigen, dass es zu einer monotonen Funktion<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf einem<\/span> <span class=\"ecti-1095\">Intervall <\/span><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 class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens abz<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlbar viele<\/span> <span class=\"ecti-1095\">Punkte geben kann, bei denen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nicht stetig ist (sogenannte Unstetigkeitsstellen). Gehen Sie dazu wie folgt vor: Sei<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">die Menge der<\/span> <span class=\"ecti-1095\">Unstetigkeitsstellen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/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\">Sei <\/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> <span class=\"ecti-1095\">Wir setzen<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>x<\/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\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">W<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie anschliessend eine rationale Zahl<\/span> <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Wir nehmen hier an, dass wir beliebig oft eine Wahl treffen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.)<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">injektiv ist und schliessen Sie auf die Aussage.<\/span><\/dd><\/dl> <\/div> <div class=\"wp-nocaption \"><\/div> \n","protected":false},"author":1089,"menu_order":9,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-50","chapter","type-chapter","status-publish","hentry"],"part":41,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/50","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\/50\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/41"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/50\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=50"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=50"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=50"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}