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zm<>^{Qt-%F0kRFm`2q#BV-f2TKm!IQUkVh^5i|f0K?B(of(D9>)iiLJf`i*2pdoAl zBv)v7-U(=nXT2+R1A`?a`XCrk=E37226Uz1XIX%TIA;QfV8Hren1LII>o|sp1`!xG zjtF8N{8|v>;XOPI5u69WcmU{oL8W&xAI)W#mUeTyn+tA!+;B=W_iwZ@!%eH(%#+$`~~nq zchnxYGqVT3!0d~^q`7nsOUX)LNjO;ojwm68K}*VzrHB$F0!E4?AtMViAEFr7e;%U8 b1#ZK2hVEWnyk`Il*a;q2;^UK1mZkZB`Huiw literal 0 HcmV?d00001 diff --git a/ana4.tex b/ana4.tex new file mode 100644 index 0000000..88dabc5 --- /dev/null +++ b/ana4.tex @@ -0,0 +1,240 @@ +\documentclass{lecture} + +\begin{document} +\newcommand{\icol}[1]{% inline column vector + \left(\begin{smallmatrix}#1\end{smallmatrix}\right)% +} +\newcommand{\K}{\mathbb{K}} + +\chapter{Der $n$-dimensionale Zahlenraum $K^{n}$} + +\section{Der euklidische Raum $K^{n}$} + +\begin{bem} + $K^{n}$ bezeichnet den Vektorraum der $n$-Tupel $x = \icol{x_{1} \\ \vdots \\ x_{n} }, x_{i} \in \K, i = 1,...,n, \\ n \in \N$ + mit Addition $x+y \coloneqq \icol{x_{1} + y_{1} \\ \vdots \\ x_{n}+y_{n} }$ und skalarer Multiplikation $\alpha \cdot x = \icol{\alpha x_{1} \\ \vdots \\ \alpha x_{n} }, \forall \alpha \in \K$. +\end{bem} + +\begin{definition} + Sei $X$ irgendeine Menge. \\ + Eine Metrik auf $X$ ist eine Abbildung $d: X \times X \to \R, \ (x,y) \to d(x,y)$ mit folgenden Eigenschaften: + \begin{enumerate}[M1] + \item (Definitheit) $d(x,y) \geq 0$, $d(x,y)=0 \Leftrightarrow x=y$ + + \item (Symmetrie) $\forall x,y \in X$ gilt $d(x,y) = d(y,x)$. + + \item (Dreiecksungleichung) $\forall x,y,z \in X$ gilt $d(x,z) \leq d(x,y) + d(y,z)$. + \end{enumerate} + Ein metrischer Raum $(X, d)$ besteht aus einer Menge $X$ mit einer Metrik $d$. Man nennt $d(x,y)$ auch den Abstand oder die Distanz von $x$ und $y$. +\end{definition} + +\begin{bsp} + \begin{enumerate}[(1)] + \item $X = \R$ oder $\C$ mit $d(x,y) \coloneqq |x-y|$ ist ein metrischer Raum, denn: + \begin{enumerate}[M1] + \item folgt aus $|x| = 0 \Leftrightarrow x=0$, $|x| \geq 0$, + \item folgt aus $|x-y| = |-(y-x)| = |y-x|$, + \item folgt aus $d(x,y) = |x-y| = |x-z+z-y| \leq |x-z| + |z-y| = d(x,z) + d(z,y)$. + \end{enumerate} + + \item (induzierte Metrik) Sei $(X,d)$ metrischer Raum und $A \subseteq X$. Die induzierte Metrik $d_{A}$ auf $A$ ist definiert durch $d_{A}: A \times A \to \R, \ d_{A}(x,y) \coloneqq d(x,y), \ \forall x,y \in A$. Dann wird $(A, d_{A})$ zu einem metrischen Raum. + + \item (triviale Metrik) Sei $X$ eine Menge. Die Triviale Metrik wird definiert durch: $$ d(x,y) = \begin{cases} 0, & \text{für} \ x=y \\ 1, & \text{für} \ x \neq y \end{cases}.$$ + + \item Weiteres wichtiges Beispiel: metrische Räume entstehen aus normierten Vektorräumen. + \end{enumerate} +\end{bsp} + +\begin{definition} + (normierter Raum) Sei $V$ irgendein Vektorraum über $\K$ ($\K = \R$ oder $\K = \C$). Eine Abbildung $\norm{\cdot}: V \to \R$ heißt Norm (auf V), wenn folgende Bedingungen erfüllt sind: + \begin{enumerate}[N1] + \item (Definitheit) $\norm{x} \geq 0$, $\norm{x} = 0 \Leftrightarrow x=0$. + \item (Homogenität) $\norm{ \alpha x} = |\alpha| \cdot \norm{x}, \alpha \in \K$. + \item (Dreiecksungleichung) $\norm{x+y} \leq \norm{x} + \norm{y}$. + \end{enumerate} + Das Paar $(V, \norm{\cdot})$ heißt normierter Raum. +\end{definition} + +\begin{bem} + (Norm $\leadsto$ Metrik) Sei $(V, \norm{\cdot})$ ein normierter Raum. Dann ist $d(x,y) \coloneqq \norm{x-y}, \forall x,y \in V$ eine Metrik auf $V$. +\end{bem} + +\begin{bsp} + Normen in $\R^{n}$. \begin{enumerate}[(1)] + \item Euklidische Norm $\norm{x}_{2} \coloneqq \sqrt{\sum_{i=1}^{n} x_{i}^{2}}$. + \item Maximumsnorm oder $\ell^{\infty}$-Norm: $\norm{x}_{\infty} \coloneqq \max_{i = 1,..,n} |x_{i}|$. + \item $\ell^{1}$-Norm: $\norm{x}_{1} \coloneqq \sum_{i=1}^{n} |x_{i}|$. + \item $\ell^{p}$-Norm: $\norm{x}_{p} \coloneqq \sqrt[p]{\sum_{i=1}^{n} |x_{i}|^{p}}$. + \end{enumerate} +\end{bsp} + +\begin{definition} + Eine Folge $(x^{(k)})_{k \in \N}, x^{(k)} \in \K$, heißt + \begin{enumerate}[i)] + \item beschränkt, falls $\forall x^{(k)} \in K_{R}(0)$, $K_{R}(0)$ eine Kugelumgebung von $0$ mit Radius $R$. + $$K_{r}(0) \coloneqq \{ x \in \K^{n} \ | \ \norm{x-a}_{\infty} < r\}.$$ + \item Cauchy-Folge, wenn $\forall \varepsilon > 0, \exists N_{\varepsilon} \in \N$ sodass $\forall k,l \geq N_{\varepsilon}$ gilt: $\norm{x^{(k)} - x^{(l)}}_{\infty} < \varepsilon$. + \item konvergent gegen ein $x \in \K^{n}$, wenn $\norm{x^{(k)} - x}_{\infty} \to 0$ für $k \to \infty$. \\ + geometrisch: jede Kugelumgebung $K_{\varepsilon}(x)$ enthält fast alle Folgenelemente $x^{(k)}$ enthält (d.h. alle bis auf endlich viele). + \end{enumerate} +\end{definition} + +\begin{bem} + Offenbar: + \begin{align*} + &\norm{x^{(k)} - x}_{\infty} \to 0, k \to \infty & & \Leftrightarrow \\ + &\left| x_{i}^{(k)} - x_{i} \right| \overset{k \to \infty}{\longrightarrow} 0, i = 1,...,n. + \end{align*} + Das heißt $\lim_{n \to \infty} x_{i}^{(k)} = x_{i}$ (komponentenweise Konvergenz in $\R$ oder $\C$) +\end{bem} + +\begin{satz} + (Satz von Cauchy und Satz von Bolzano-Weierstraß) + \begin{enumerate}[1)] + \item Jede Cauchy-Folge in $K^{n}$ konvergiert, d.h. der normierte Raum $(K^{n}, \norm{\cdot}_{\infty})$ ist vollständig. Ein vollständiger normierter Raum wird Banach-Raum genannt. + \item Jede beschränkte Folge in $K^{n}$ besitzt eine konvergente Teilfolge. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[1)] + \item Sei $(x^{(k)})_{k \in \N}$ eine Cauchy-Folge, d.h. $\forall \varepsilon > 0, \exists N_{\varepsilon}, \forall k,l \geq N_{\varepsilon}$ gilt $\norm{x^{(k)} - x^{(l)}}_{\infty} < \varepsilon$. + Betrachte Komponentenfolge $(x_{i}^{(k)})_{k \in \N}, i = 1,...,n$. Die Komponentenfolgen sind Cauchy-Folgen, weil $$\left| x_{i}^{(k)} - x_{i}^{(l)} \right| \leq \norm{x^{(k)} - x^{(l)}}_{\infty} < \varepsilon, \forall k,l \geq N_{\varepsilon}, \forall i = 1,...,n.$$ + $\implies \lim_{k \to \infty} x_{i}^{(k)} \eqqcolon x_{i} \implies x^{(k)} \overset{k \to \infty}{\longrightarrow} x = \icol{x_{1} \\ \vdots \\ x_{n}}$ in $\ell_{\infty}$ Norm. + \item Sei $(x^{(k)})_{k \in \N}$ beschränkt $\implies$ $(x_{i}^{(k)})_{k \in \N}, \forall i = 1,...,n$ auch beschränkt \\ + $\stackrel{\text{Bo.-We. in} \ \K}{\implies}$\ \ \ \ es existiert eine konvergente Teilfolge $(x_{1}^{(k_{1,j})})_{j \in \N}$ von $(x_{1}^{(k)})_{k \in \N}$ mit $x_{1}^{(k_{1,j})} \overset{j \to \infty}{\longrightarrow} x_{1}$, \\ + nach $n$ Schritten haben wir Teilfolgen $(x_{n}^{(k_{n,j})})_{j \in \N}$ von $(x_{n}^{(k)})_{k \in \N}$ für die alle Komponentenfolgen konvergieren $(x_{i}^{(k_{i,j})})_{j \in \N} \overset{j \to \infty}{\longrightarrow} x_{i}, \forall i = 1,...,n$. Daraus folgt $(x^{k_{nj}}) \overset{j \to \infty}{\longrightarrow} x$. + \end{enumerate} +\end{proof} + +\begin{satz} + (Äquivalenz von Normen) Sei $\K^{n}$ ein endlichdimensionaler Vektorraum. Dann sind alle Normen äquivalent zur Maximumsnorm $(\ell_{\infty})$, d.h. zu jeder Norm $\norm{\cdot}, \exists m,M >0$ sodass $$m \norm{x}_{\infty} \leq \norm{x} \leq M \norm{x}_{\infty}, \ \ \ \ x \in \K^{n}.$$ +\end{satz} + +\begin{proof} + Sei $\norm{\cdot}$ irgendeine Norm. $\forall x \in \K^{n}, x = \sum_{k=1}^{n} x_{k}e^{(k)}$, wobei $e^{(k)}, k=1,...,n$ die sogenannte euklidische Basis ist: $e^{(k)} = \icol{\delta_{k,1} \\ \vdots \\ \delta_{k,n} }$. \\ + Dann: + \begin{align*} + \norm{x} & \leq \sum_{k=1}^{n} |x_{k}|\cdot \norm{e^{(k)}} \\ + & \leq \sum_{k=1}^{n} \max_{i = 1...n} |x_{i}| \cdot \norm{e^{(k)}} \leq M \cdot \norm{x}_{\infty}. + \end{align*} + Wobei $M \coloneqq \sum_{k=1}^{n} \norm{e^{(k)}}$. \\ Setze $$S_{1} \coloneqq \{ x \in \K^{n} \ | \ \norm{x}_{\infty} = 1\}, \ \ m \coloneqq \inf \{ \norm{x} \ | \ x \in S_{1} \} \geq 0.$$ \\ + Es gzz.: $m > 0$. Annahme $m=0$. Dann existiert eine Folge $(x^{(k)})_{k \in \N}$, $x^{(k)} \in S_{1}$, sodass $\norm{ x^{(k)} } \overset{k \to \infty}{\longrightarrow} 0$. Aus $x^{(k)} \in S_{1}$ folgt $(x^{(k)})_{k \in \N}$ ist beschränkt in der $\ell_{\infty}$-Norm. Dann impliziert der Satz von Bolzano-Weierstraß: es existiert eine konvergente Teilfolge, o.B.d.A. $(x^{(k)}) \overset{k \to \infty}{\longrightarrow} x$ in der $\ell_{\infty}$-Norm, dann: + \begin{align*} + & \ \ \ \underbrace{\left| \underbrace{\norm{x^{(k)}}_{\infty}}_{=1} - \norm{x}_{\infty} \right|}_{=|1-\norm{x}_{\infty}} \leq \underbrace{\norm{x^{(k)} -x}_{\infty}}_{\overset{k \to \infty}{\longrightarrow} 0} + \implies |1-\norm{x}_{\infty} \overset{k \to \infty}{\longrightarrow} 0 + \implies \norm{x}_{\infty} = 1 \implies x\in S_{1}. + \end{align*} + Anderseits: + \begin{align*} + &\norm{x} \leq \norm{x-x^{(k)}} + \norm{x^{(k)}} \leq M \cdot \norm{x-x^{(k)}}_{\infty} + \norm{x^{(k)}} \overset{k \to \infty}{\longrightarrow} 0 \\ + \implies \ & \ \norm{x}=0 \ \ \implies x=0. + \end{align*} + Widerspruch zu $x \in S_{1}$, also $m>0$. Dann für $x \neq 0$ ist Vektor $\frac{x \ }{\norm{x}_{\infty}} \in S_{1}$ und $m \leq \frac{\norm{x} \ }{\norm{x}_{\infty}}$ (nach Definition von $m$) und $0 < m \cdot \norm{x}_{\infty} \leq \norm{x}, \ x \in \K^{n}$. +\end{proof} + +\begin{korrolar} + Auf $K^{n}$ sind alle Komvergenzen in irgendeiner Norm äquivalent zur Konvergenz in der $\ell_{\infty}$-Norm. (= komponentenweiser Konvergenz) +\end{korrolar} + +\begin{bem} + Obiger Satz gilt nicht für unendlich dimensionale Räume (wie z.B. $C[a,b], R[a,b]$). Die endliche Dimension von $K^{n}$ ist entscheidend. +\end{bem} + + +\section{Teilmengen in $K^{n}$ (Topologische Grundbegriffe)} + +Bezeichnung: $\norm{\cdot}$ irgendeine Norm. + +\begin{definition} + ($\varepsilon$-Kugel, $\varepsilon$-Umgebung) Sei $a \in \K^{n}, r>0$. + \begin{enumerate}[(1)] + \item Dann heißt $K_{r}(a) \coloneqq \{ x \in \K^{n} \ | \ \norm{a-x} < r \}$ die offene Kugel um $a$ mit Radius $r$ bzgl. $\norm{\cdot}$. + \item $U \subset K^{n}$ heißt Umgebung von $a \in \K^{n}$, falls $\exists \varepsilon > 0$ mit $K_{\varepsilon}(a) \subseteq U$. Insbesondere ist $K_{\varepsilon}(a)$ selbst eine Umgebung von $a$, eine sogenannte $\varepsilon$-Umgebung von $a$. + \end{enumerate} +\end{definition} + +\begin{definition} + (offene Menge) Eine Menge $O \in \K^{n}$ heißt offen, falls $O$ eine Umgebung jedes Punktes aus $O$ ($ x \in O$) ist, das heißt $\forall x \in O, \exists \varepsilon > 0$ mit $K_{\varepsilon}(x) \subseteq O$. +\end{definition} + +\begin{bsp} + \begin{enumerate}[(1)] + \item $]a,b[ \subseteq \R$ ist offen ($a 0$, die $a 0$ mit $K_{\varepsilon_{1}}(x) \subseteq U$, $K_{\varepsilon_{2}}(x) \subseteq V$. Damit gilt für $\varepsilon \coloneqq \min\{ \varepsilon_{1}, \varepsilon_{2} \}$, $K_{\varepsilon}(x) \subseteq U \cap V$. (Beachte: $\emptyset$ ist immer offen.) + \item Sei $x \in \underset{i \in I}{\bigcup} U_{i}$, dann $\exists j \in I$ mit $x \in U_{j}$. \\ $U_{j}$ offen $\implies$ $\exists \varepsilon > 0$ mit $K_{\varepsilon}(x) \subseteq U_{j}$ $\implies$ $K_{\varepsilon}(x) \subseteq \underset{i \in I}{\bigcup} U_{i}$. + \end{enumerate} +\end{proof} + +\begin{korrolar} + \begin{enumerate}[1)] + \item Endliche Schnitte und beliebige Vereinigung von offenen Mengen sind wieder offen. + \item (Beobachtung) Durchschnitt von unendlich vielen offenen Mengen braucht nicht offen zu sein. Z.B. $$ \overset{\infty}{\underset{n=1}{\bigcap}} ]-\frac{1}{n}, 1 + \frac{1}{n}[ = [0,1]$$ ist nicht offen, da $K_{\varepsilon}(0) \subset [0,1], \forall \varepsilon > 0$. + \end{enumerate} +\end{korrolar} + +\begin{definition} + (Abgeschlossene Menge) Eine Teilmenge $A \subset \K^{n}$ heißt abgeschlossen, wenn ihr Komplement $A^{c} \coloneqq \K^{n} \setminus A$ offen ist. +\end{definition} + +\begin{bsp} + \begin{enumerate}[(1)] + \item Für $a,b \in \R, a \leq b$ ist $[a,b]$ abgeschlossen, denn $]-\infty,a[ \cup ]b, \infty[ = \R \setminus [a,b]$ ist offen, denn: + \begin{align*} + ]-\infty, a[ & = \underset{n \in \N}{\bigcup} ]a-n,a[ \ \ \text{ist offen} & + ]b, \infty[ = \underset{n \in \N}{\bigcup} ]b, b + n[ \ \ \text{ist offen}. + \end{align*} + \end{enumerate} +\end{bsp} + +\begin{satz} + (Eigenschaften abgeschlossener Mengen) + \begin{enumerate}[(1)] + \item Sind $V, U$ ($V,U \subset \K^{n}$) abgeschlossen, dann ist $U \cup V \subset \K^{n}$ auch abgeschlossen. + \item Sind $U_{i}, (i \in I)$ abgeschlossene Menge in $\K^{n}$. Dann ist $\underset{i \in I}{\bigcap} U_{i}$ auch abgeschlossen. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[(1)] + \item $(U \subset V)^{c} = \underbrace{U^{c}}_{\text{offen}} \cap \underbrace{V^{c}}_{\text{offen}}$ offen. + \item $\left( \underset{i \in I}{\bigcap} U_{i} \right)^{c} = \underset{i \in I}{\bigcup} \underbrace{U_{i}^{c}}_{\text{offen}}$ offen. + \end{enumerate} +\end{proof} + +\begin{bsp} + \begin{enumerate}[(1)] + \item Beliebige Vereinigung abgeschlossener Mengen muss nicht abgeschlossen sein. Z.B. $]0,1[ = \underset{n \in \N}{\bigcup} \underbrace{[\frac{1}{n}, 1- \frac{1}{n}]}_{\text{abgeschlossen}}$ ist offen. + \item $\emptyset$ und $K^{n}$ sind abgeschlossen. + \item $A_{1} \subset \R^{n_{1}}$ und $A_{2} \subset \R^{n_{2}}$ abgeschlossen, dann ist auch $A_{1} \times A_{2} \subset \R^{n_{1}} \times \R^{n_{2}} = \R^{n_{1} + n_{2}}$ abgeschlossen. + \item Für $a?hxD)+}%A89D+;G;11s;+57D5 zd(OG%e($~WXPAfXuGL*tt5?+knXHfqEd!7Ro^1R}&n!GMfF59_X9~~71)x&^(1}_p zS^?;U0B`h-b*=5ewm=3-01pp5*wWy~Al5$yzY(!AHv(G%94rk0@?aBV$d8UzmH-h4 zOEY^DD@(8?<)1Uy{>}1sVErE>fee2)z`*bi3;$Tl@Go0N=3oN;2e1l4tE^fdok8}uL7X8POO z?0>E;ZfOrTveo^u>!0T)Y74=v!Nkg}2Z8@%N0vt5f0$vU|Bq&ve>cO(^rxA>Z-G|+ z#~$SWz6ZvC=kvdu!t%Gpe{;KotqItcR@le2l<{GG$@+XgXC)l>b0Na#g)pe#w?Qzk zbdC?W4eYH=9+~f3^Fytv;C6{V$@=;)Ew>^K=ndh>Mo7j8I6ny$eaj82j^2l2X)RY$ z>)H}%INlqr;AX)VdX9?}J?ahH0xMH(6Q&v_f}L*1q1oFalUvAxki@0xTWO%{70Y`Dgt7@Y#Qz$@p_7Gd++E#Q4uM|B7^Y0Az3GR<;V( zy82+ik0^iyhpetW#M_pDpF0E42|L@1D%e9D5BL>4kZ^z;2jB;zi!g!!%s-<0M@TWU 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