From a8d317235c61ba5abd952d8095d81a4337e8dedb Mon Sep 17 00:00:00 2001 From: salagne Date: Wed, 10 Jun 2020 14:53:52 +0200 Subject: [PATCH] corrections --- ana1.pdf | Bin 317567 -> 244287 bytes ana1.tex | 12 ++++++------ ana10.pdf | Bin 145988 -> 145988 bytes ana11.pdf | Bin 168424 -> 168424 bytes ana13.pdf | Bin 180135 -> 159962 bytes ana13.tex | 34 +++++++++++++++++----------------- ana2.pdf | Bin 229289 -> 172650 bytes ana2.tex | 20 ++++++++++---------- ana3.pdf | Bin 225988 -> 200285 bytes ana4.pdf | Bin 265330 -> 196095 bytes ana4.tex | 6 +++--- ana5.pdf | Bin 191984 -> 146839 bytes ana5.tex | 12 ++++++------ ana6.pdf | Bin 273418 -> 194548 bytes ana7.pdf | Bin 296633 -> 220317 bytes ana7.tex | 12 +++++++----- analysisII.pdf | Bin 618037 -> 618110 bytes analysisII.tex | 2 +- lecture.cls | 1 + 19 files changed, 51 insertions(+), 48 deletions(-) diff --git a/ana1.pdf b/ana1.pdf index f9831dfa475829faef8f01d54b299740d5a7a095..b9ffa93b40fae596174bd56ad8f21be829e09b0d 100644 GIT binary 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(oder $\mathbb{C}$) stetig. Dann \[ - \Vert f \Vert_\infty := \max \{ |f(x)| \mid x \in [a,b]\} + \Vert f \Vert_\infty \coloneqq \max \{ |f(x)| \mid x \in [a,b]\} .\] Das Maximum existiert wegen der Stetigkeit des Absolutbetrags und weil $[a,b]$ beschränkt und abgeschlossen ist. \end{definition} @@ -240,7 +240,7 @@ \forall n,m \ge N_0 \quad \forall x \in [a,b] .\] $\implies (f_n(x))_{n\in\N}$ ist eine Cauchy-Folge in $\R$.\\ $\implies (f_n(x))_{n\in\N}$ konvergiert \\ - $\implies$ Definiere $f(x) := \lim_{n \to \infty} f_n(x)$, $x \in [a,b]$. + $\implies$ Definiere $f(x) \coloneqq \lim_{n \to \infty} f_n(x)$, $x \in [a,b]$. Dann gilt $\forall n \ge N_0$, $\forall x \in [a,b]$: \[ @@ -262,7 +262,7 @@ \begin{definition} Der Funktionenraum $C[a,b]$ definiert durch \[ - C[a,b] := \{ f \colon [a,b] \to \R \mid f \text{ stetig }\} + C[a,b] \coloneqq \{ f \colon [a,b] \to \R \mid f \text{ stetig }\} ,\] ist mit $\Vert f \Vert_\infty$ ein normierter Vektorraum. \end{definition} @@ -306,7 +306,7 @@ Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^ $\sum_{n=0}^{\infty} f_n$ konvergiere gleichmäßig auf $[a,b]$, d.h. die Folge der Partialsummen $(\sum_{n=0}^{N} f_n)_{n\in\N}$ sei gleichmäßig konvergent. 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Dann gilt: + Seien $D \subset \R^{n}$ offen, $f: D \to \R^{m}$ stetig differenzierbar. Sei $x \in D$ und $\varepsilon > 0$ sodass $K_{\varepsilon}(x) \subset D$. Dann gilt: \begin{salign*} \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall y \in K_{\varepsilon} \end{salign*} mit $M \coloneqq \sup_{z \in K_{\varepsilon}(x)} \norm{J_{f}(z)}_{2}$, das heißt $f$ ist lokal Lipschitz-stetig. \\ Sei $D$ konvex, dann gilt \begin{salign*} - \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall x,y \in K_{\varepsilon} + \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall x,y \in D \end{salign*} mit $M \coloneqq \sup_{z \in D} \norm{J_{f}(z)}_{2}$, das heißt $f$ ist auf $D$ Lipschitz-stetig. \end{korrolar} \begin{proof} Aus obigem Lemma folgt: \begin{salign*} - \norm{\int_{0}^{2} J_{f}(x+sh) h \d{s} }_{2} &\leq \int_{0}^{1} \norm{J_{f}(x+sh)}_{2} \norm{h}_{2} \d{s} \\ & \leq \int_{0}^{1} \norm{J_{f}(x+sh)}_{2} \norm{h}_{2} \d{s} \leq \sup_{0*l)c*1>H) zcIaDAfH5+Hsn^z#JgP9foJOLH;%cU(Uiuy`U!TO&(>9}{)xFy=bHUjVPmh>AL*gxxgj-3Q>Fz`(g+o-2_}09 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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) $\norm{x+y} \leq \norm{x} + \norm{y}$. + \item (Definitheit) $\norm{x} \geq 0$, $\norm{x} = 0 \Leftrightarrow x=0$. \label{def:N1} + \item (Homogenität) $\norm{ \alpha x} = |\alpha| \cdot \norm{x}, \alpha \in \K$. \label{def:N2} + \item (Dreiecksungleichung) $\norm{x+y} \leq \norm{x} + \norm{y}$. \label{def:N3} \end{enumerate} Das Paar $(V, \norm{\cdot})$ heißt normierter Raum. \end{definition} diff --git a/ana5.pdf b/ana5.pdf index b0c0e94e536b0daa7067a648911fc798336ceb4e..d4d8a7545936d7a059e80014334c6ad12e3f678a 100644 GIT binary patch delta 52503 zcmZs?Lv$`&w5}Q3wr$(ClO5YTw!hf6ZQHhYY}+>LKjTz2YTVnKjn#O1bG?{%o3Lr0 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