From 04bdda2272d595ca9df5fb78c740d82cec7fb99e Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 19 May 2020 20:14:31 +0200 Subject: [PATCH] add ana8, one proof in ana7 and replace epsilon with varepsilon globally --- ana7.pdf | Bin 235931 -> 220501 bytes ana7.tex | 47 ++++++++- ana8.tex | 257 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 578318 -> 522670 bytes analysisII.tex | 1 + lecture.cls | 3 + 6 files changed, 307 insertions(+), 1 deletion(-) create mode 100644 ana8.tex diff --git a/ana7.pdf b/ana7.pdf index 2b28862dfa577e3b51d36b309033d53a4907bf41..148fad22e54157b9d1116b5f30c702c2685c01df 100644 GIT binary patch delta 69225 zcmZs?V{I+qR9B{k;2p+^4H%RsVpl>KZ++nbVB0 z(glx5t|TTw&&&&HI}}DW7Nk z$Ad>Sg7Skd+D5v{Irp^R6LSDRh;)L~eDXCyOA;j_WAwIpjse5_{yXAM+r5*oXvJK- zno)7Kpn%Vd#{DF6Nl$Z*k!Cm&t%+q4rP}+&Pp214d1&H7%a;H1wc1jV;OG9>jV=_A zpU1r9>8T#j=l^}w%viu@+kfQoe5CqKB#}_y=lQwxR8PofPKO<&DiMpp)OYutvmUCE z%|%Mb;T(#s4PBfC6PG_KL30&>0y;A$%^QBxwes>S;5Df}N`{Q0>7g7)9#T1QJQj>n 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z!?XYhhcO6hz$6x6Nn|XDq<={7SiD0!4Etd(0o!a+g541t!@d{n(HKtxTZ|-}rXiXE znxGVsc!!&1@CWG(+h%AwmNGu7YM`-biiX_)Y?TBwMjx_<`j3PlK^+8W*nY0`Yu{+$ zNpd79U8OSiBUsVEJ{(44P0Me-^^v2m|A2_?WWxFu z$jE>cBzTlTD9F);CIvx^Ixx^6vzd-30ulC4Oo>3l@&V8wk74Z~CQurj^B^XnAbD1m zT-s^~Xs}OiRR3wzAyF|ZwR!jM3Map)2`c%R)5$kJ9 0}) \Vert x \Vert + .\end{align*} + Also hat die Gleichung $(\mathbb{I} + B) x = 0$ nur die Lösung $x = 0$, also + ist $(\mathbb{I} + B)$ injektiv und mit \ref{lemma:linabb} regulär. + + Bleibt zu zeigen: $\Vert (\mathbb{I} + B)^{-1} \Vert \le \frac{1}{1 - \Vert B \Vert}$. + Es gilt + \begin{align*} + 1 \qquad &= \qquad \Vert \mathbb{I}\Vert \\ + &= \qquad \Vert (\mathbb{I} + B) (\mathbb{I} + B)^{-1} \Vert \\ + &= \qquad \Vert (\mathbb{I} + B)^{-1} + B (\mathbb{I} + B)^{-1} \Vert \\ + &\stackrel{\text{Dreicksungl.}}{\ge } \qquad \Vert (\mathbb{I} + B)^{-1} \Vert + - \Vert B (\mathbb{I} + B)^{-1} \Vert \\ + &\ge \qquad \Vert (\mathbb{I} + B)^{-1} \Vert - \Vert B \Vert \cdot \Vert (\mathbb{I} + B)^{-1} \Vert \\ + &= \qquad (1 - \Vert B \Vert) \Vert (\mathbb{I} + B)^{-1} \Vert + .\end{align*} + Damit folgt die Behauptung. +\end{proof} + +\begin{korrolar} + Sei $A \in \mathbb{K}^{n \times n}$ regulär und + $\tilde A \in \mathbb{K}^{n \times n}$ s.d. $\Vert A - \tilde A\Vert < \frac{1}{\Vert A^{-1} \Vert}$. Dann + ist $\tilde A$ regulär. +\end{korrolar} + +\begin{proof} + Es ist $\tilde A = \tilde A + A - A = (\tilde A - A) + A = A + (\underbrace{A^{-1} (\tilde A - A) + \mathbb{I}}_{=:B})$. Damit folgt + $\Vert B \Vert = \Vert A^{-1} (\tilde A - A) \Vert \le \Vert A^{-1} \Vert \cdot \Vert \tilde A + - A \Vert < 1$. + Mit \ref{lemma:stoerung} folgt $\mathbb{I} + A^{-1}(\tilde A - A)$ regulär. Da + A regulär nach Vorraussetzung, folgt $\tilde A = A (\mathbb{I} + A^{-1} (\tilde A - A))$ regulär. +\end{proof} + +\chapter{Funktionen mehrerer Variablen} + +Wir betrachten im Folgenden Funktionen $f\colon D \to \mathbb{K}$, mit +$D \subseteq \mathbb{K}^{n}$, $D \neq \emptyset$ und Bildbereich $B_f \subseteq \mathbb{K}$. + +Zur Erinnerung: +\begin{itemize} + \item \underline{Bild und Urbild}. Seien $M \subseteq D$, $N \subseteq f(D)$ + Teilmengen. Dann heißt + \begin{align*} + f(M) &:= \{y \in \mathbb{K} \mid \exists x \in M\colon y = f(x)\} + \intertext{das Bild. Weiter heißt} + f^{-1}(N) &:= \{ x \in D \mid \exists y \in N\colon f(x) = y\} + .\end{align*} + das Urbild. Dann ist $B_f = f(D)$ und $D = f^{-1}(B_f)$ + \item \underline{Notation}. $f^{-1}(\cdot )$ meint das Mengen-Urbild, \underline{nicht} + eine Umkehrfunktion. +\end{itemize} + +Da alle Normen auf $\mathbb{K}^{n}$ äquivalent sind, sind alle Aussagen unabhängig +von der gewählten Norm. Standard ist die euklid. Norm. + +\section{Stetigkeit} + +\begin{definition}[Stetigkeit] + Eine Funktion $f\colon D \to \mathbb{K}$, $D \subseteq \mathbb{K}^{n}$ heißt + \underline{stetig} in einem Punkt $a \in D$, wenn für alle Folgen + $\left( x^{(k)} \right)_{k \in \N} \subseteq D$ mit + $x^{(k)} \xrightarrow{k \to \infty} a$ gilt + \[ + f\left( x^{(k)} \right) \xrightarrow{k \to \infty} f(a) + .\] Die Funktion $f$ heißt \underline{stetig in $D$}, wenn sie für alle + $x \in D$ stetig ist. +\end{definition} + +\begin{bem} + \begin{itemize} + \item Falls $f\colon D \to \mathbb{K}$ stetig, dann ist auch + $f\colon M \to \mathbb{K}$, $M \subseteq D$ stetig. + \item $f$ stetig $\implies \text{Re } f$, $\text{Im } f$, $|f|$ sind stetig. + \end{itemize} +\end{bem} + +\begin{lemma}[$\varepsilon$-$\delta$-Kriterium der Stetigkeit] + $f\colon D \to \mathbb{K}$, $D \subseteq \mathbb{K}^{n}$ ist stetig in $a \in D$, genau + dann wenn $\forall \epsilon > 0$, $\exists \delta > 0$, s.d. $\forall x \in D$ gilt + \[ + \Vert x - a \Vert < \delta \implies |f(x) - f(a)| < \epsilon + .\] +\end{lemma} + +\begin{proof} + wie für $n = 1$. +\end{proof} + +\begin{lemma} + Seien $f, g\colon D \to \mathbb{K}$ stetig, dann sind + $f + g$, $f \cdot g$ und $\frac{f}{g}$ (falls $g(x) \neq 0$ $\forall x \in D$) + stetig. +\end{lemma} + +\begin{proof} + wie für $n = 1$. +\end{proof} + +\begin{satz} + Eine stetige Funktion $f\colon D \to \mathbb{K}$, $D \subseteq \mathbb{K}^{n}$ + ist auf jeder kompakten Menge $K \subseteq D$ beschränkt, d.h. + \[ + \exists M_K \text{ s.d. } |f(x)| \le M_K \quad \forall x \in K + .\] +\end{satz} + +\begin{proof} + Ang.: $f(x)$ nicht beschränkt auf $K$. Dann gilt: $\forall k \in \N$, $\exists x^{(k)} \in K$ mit + $|f\left(x^{(k)}\right)| > k$, d.h. $|f\left( x^{(k)} \right)| \xrightarrow{k \to \infty} \infty$. + + Die Folge $(x^{(k)})_{k\in\N}$ besitzt auf der kompakten Menge $K$ eine + konvergente Teilfolge $\left( x^{(k_j)} \right)_{j \in \N}$ mit + $\displaystyle \lim_{j \to \infty} x^{(k_j)} = x \in K$. + + Da $f$ stetig, folgt $|f\left( x^{(k_j)} \right)| \xrightarrow{j \to \infty} |f(x)|$. Widerspruch + zu $|f\left( x^{(k)} \right)| \xrightarrow{k \to \infty} \infty$. +\end{proof} + +\begin{satz}[Extremum] + Eine stetige Funktion $f\colon D \subseteq \mathbb{K}^{n} \to \mathbb{K}$ nimmt + auf jeder nichtleeren kompakten Menge $K \subseteq D$ ihr + Maximum und Minimum an, d.h. es ex. $x^{max}$ und $x^{min} \in K$, s.d. + \begin{align*} + f(x^{max}) &= \sup_{x \in K} f(x) =: \max_{x \in K} f(x) \\ + f(x^{min}) &= \inf_{x \in K} f(x) =: \min_{x \in K} f(x) + .\end{align*} + \label{satz:stetigextremum} +\end{satz} + +\begin{proof} + $f$ stetig und deshalb beschränkt auf $K$, d.h. es ex. obere Schranke + $\displaystyle M := \sup_{x \in K} f(x)$. Außerdem existiert eine Folge + $\left( x^{(k)} \right)_{k \in \N} \subseteq K$, s.d. + $f\left( x^{(k)} \right) \xrightarrow{k \to \infty} M$. Da $K$ kompakt, existiert + eine konvergente Teilfolge $\left( x^{(k_j)} \right)_{j \in \N}$ mit + $x^{(k_j)} \xrightarrow{j \to \infty} a =: x^{max} \in K$. Wegen der Stetigkeit von + $f$, folgt aus $f\left( x^{(k_j)} \right) \xrightarrow{j \to \infty} f\left( x^{max} \right)$: + $f(x^{max}) = M$. +\end{proof} + +\begin{bem}[Anwendung von Satz \ref{satz:stetigextremum}] + Seien $K_1, K_2 \subseteq \mathbb{K}^{n}$, $K_1 \neq \emptyset$, $K_2 \neq \emptyset$ + kompakt. Dann ist die Menge $K_1 \times K_2$ auch kompakt. Definiere + $f(x, y) := \Vert x - y \Vert$, $x \in K_1$, $y \in K_2$. + + $f(x,y)$ ist stetig, denn + \[ + |f(x,y) - f(x', y')| = | \Vert x - y \Vert - \Vert x' - y'\Vert | + \quad \stackrel{\Delta -\text{ungl.}}{\le} \quad \Vert x - y - x' + y' \Vert \le \Vert x - x' \Vert + \Vert y - y' \Vert + .\] $\forall x, x' \in K_1$ und $\forall y, y' \in K_2$ mit + $\Vert x - x'\Vert < \delta = \frac{\epsilon}{2}$ und $\Vert y - y'\Vert < \delta = \frac{\epsilon}{2}$ + gilt + \[ + |f(x,y) - f(x', y')| < \epsilon + .\] Also ist $f(x,y)$ stetig auf $K_1 \times K_2$. Mit \ref{satz:stetigextremum} folgt damit: + $\exists a \in K_1$, $b \in K_2$, s.d. + \[ + \Vert a - b \Vert = \inf_{x \in K_1 y \in K_2} \Vert x - y \Vert =: d(K_1, K_2) + \quad \text{Abstand zwischen Mengen } K_1 \text{ und } K_2 + .\] Im Fall $K_1 \cap K_2 = \emptyset$, gilt $d(K_1, K_2) > 0$. Falls + $K_1 = \{a\} $, dann heißt $b \in K_2$ die Projektion des Punktes $a$ auf + $K_2$ (diese ist im Allg. nicht eindeutig bestimmt). +\end{bem} + +\begin{definition}[Gleichmäßige Stetigkeit] + Eine Funktion $f\colon D \to \mathbb{K}$, $D \subseteq \mathbb{K}^{n}$ ist gleichmäßig stetig, + wenn $\forall \epsilon > 0$ + $\exists \delta > 0$, s.d. + \[ + \forall x, y \in D\colon \Vert x - y \Vert < \delta \implies |f(x) - f(y)| < \epsilon + .\] +\end{definition} + +\begin{satz} + Eine stetige Funktion $f\colon D \to \mathbb{K}^{n}$, $D \subseteq \mathbb{K}^{n}$ ist + auf einer kompakten Menge $K \subseteq D$ gleichmäßig stetig. +\end{satz} + +\begin{proof} + Ang. $f$ nicht gleichmäßig stetig. Dann $\exists \epsilon > 0$, s.d. $\forall k \in \N$, + ex. Punkte $x^{(k)}$ und $y^{(k)} \in K$, s.d. + \[ + \Vert x^{(k)} - y^{(k)} \Vert < \frac{1}{k} \text{ und } + \left|f\left( x^{(k)} \right) - f\left( y^{(k)} \right) \right| > \epsilon + .\] Da $K$ kompakt, ex. eine konvergente Teilfolge $\left( x^{(k_j)} \right)_{j \in \N}$ von + $(x^{(k)})_{k \in \N}$ mit $\displaystyle \lim_{j \to \infty} x^{(k_j)} = x \in K$. Wir haben + $\Vert x^{(k)} - y^{(k)} \Vert < \frac{1}{k}$, also + \[ + \Vert x^{(k_j)} - y^{(k_j)} \Vert < \frac{1}{k^{j}} \implies + \lim_{j \to \infty} y^{(k_j)} = x = \lim_{j \to \infty} x^{(k_j)} + .\] Da $f$ stetig, folgt + \[ + \left| f\left( x^{(k_j)} \right) - f\left( y^{(k_j)} \right) \right| + \xrightarrow{j \to \infty} |f(x) - f(x)| = 0 + .\] Widerspruch zu $\left| f\left( x^{(k)}\right) - f\left( y^{(k)} \right) \right| > \epsilon$. +\end{proof} + +\begin{definition}[Konvergenz von Funktionenfolgen] + Sei $f_k \colon D \subseteq \mathbb{K}^{n} \to \mathbb{K}$, $k \in \N$. + $(f_k)_{k \in \N}$ konvergiert + \begin{itemize} + \item \underline{punktweise} gegen eine Funktion $f\colon D \to \mathbb{K}$, falls + $\forall x \in D$ gilt $f_k(x) \xrightarrow{k \to \infty} f(x)$. + \item \underline{gleichmäßig}, falls $\sup_{x \in D} |f_k(x) - f(x)| \xrightarrow{k \to \infty} 0$. + \end{itemize} +\end{definition} + +\begin{satz}[Gleichmäßige Konvergenz] + Sei $f_k \colon D \subseteq \mathbb{K}^{n} \to \mathbb{K}$ stetig, $f_k \xrightarrow{k \to \infty} f$ + gleichmäßig mit $f\colon D \to \mathbb{K}$. Dann ist $f$ stetig. +\end{satz} + +\begin{proof} + Sei $x \in D$ und $\epsilon > 0$ beliebig. Da $(f_k)_{k\in\N}$ gleichmäßig gegen $f$ konvergiert, + existiert ein $n = n(\epsilon) \in \N$ s.d. + $\displaystyle \sup_{y \in D} |f_n(y) - f(y)| < \frac{\epsilon}{3}$. + + Da $f_n$ stetig, ex. $\delta > 0$, s.d. $\forall y \in D$ gilt: + $\Vert x - y \Vert < \delta \implies |f_n(x) - f_n(y)| < \frac{\epsilon}{3}$. + + Dann gilt $\forall x, y \in D$ mit $\Vert x - y \Vert < \delta $: + \[ + |f(x) - f(y)| \le |f(x) - f_n(x)| + |f_n(x) - f_n(y)| + + |f_n(y) - f(y)| < \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3} = \epsilon + .\] Also ist $f$ stetig in $x$. +\end{proof} + +\begin{bem} + Analoge Sätze gelten allgemein für Funktionen auf kompakten Mengen in + normierten $(V, \Vert \cdot \Vert)$ oder metrischen $(X, d(\cdot , \cdot ))$ Räumen. +\end{bem} + +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf index 4a50d707d05436ad83b5ff4cfdf37e73122eb15f..bace607a85902a3399e56894f047827e7c63f6d9 100644 GIT binary patch delta 293942 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-23,5 +23,6 @@ \input{ana5.tex} \input{ana6.tex} \input{ana7.tex} +\input{ana8.tex} \end{document} diff --git a/lecture.cls b/lecture.cls index ba878b5..4ee26e8 100644 --- a/lecture.cls +++ b/lecture.cls @@ -150,3 +150,6 @@ % contradiction \newcommand{\contr}{\text{\Large\lightning}} + +% people seem to prefer varepsilon over epsilon +\renewcommand{\epsilon}{\varepsilon}