From 39ed8f0f3257dc1bdebe5e16af3fada9d62adba9 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 8 Dec 2020 17:04:44 +0100 Subject: [PATCH] update algebra and ana --- ws2020/algebra/uebungen/algebra5.pdf | Bin 122252 -> 123902 bytes ws2020/algebra/uebungen/algebra5.tex | 16 ++ ws2020/ana/uebungen/ana5.pdf | Bin 0 -> 150804 bytes ws2020/ana/uebungen/ana5.tex | 215 ++++++++++++++++++++++++++ ws2020/tut-ana/loesung5.xopp | Bin 0 -> 189635 bytes ws2020/tut-ana/praesenzaufgaben5.xopp | Bin 0 -> 443462 bytes ws2020/wtheo/uebungen | 2 +- 7 files changed, 232 insertions(+), 1 deletion(-) create mode 100644 ws2020/ana/uebungen/ana5.pdf create mode 100644 ws2020/ana/uebungen/ana5.tex create mode 100644 ws2020/tut-ana/loesung5.xopp create mode 100644 ws2020/tut-ana/praesenzaufgaben5.xopp diff --git a/ws2020/algebra/uebungen/algebra5.pdf b/ws2020/algebra/uebungen/algebra5.pdf index 5bae984674de2b90a7a628dba518b89bb1200713..65eaaa477da5cafdfae2fd97606fcf3469963927 100644 GIT binary patch delta 25157 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b/ws2020/ana/uebungen/ana5.tex @@ -0,0 +1,215 @@ +\documentclass[uebung]{../../../lecture} + +\title{Analysis 3: Übungsblatt 5} +\author{Leon Burgard, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + Beh.: $f_k g_k \to fg$ in $L^{1}(X, \mu)$. + \begin{proof} + \begin{enumerate}[(i)] + \item Es ist $f$ integrabel, also $\int_{X}^{} f_+ \d{\mu} < \infty$ + und $\int_{X}^{} f_- \d{\mu} < \infty$, also + folgt $\text{ess sup } f_+ < \infty$ und $\text{ess sup } f_- < \infty$, also + folgt mit $f = f_+ - f_-$, auch $\text{ess sup } f = M < \infty$ + für ein $M \in \R$. + \item Es ist $\sup_{k \in \N} \Vert g_k \Vert_{L^{\infty}} \le S < \infty$ + für ein $S \in \R$, + also auch $\text{ess sup } g < \infty$ (Ang.: es gäbe + ein $A \in \mathcal{E}$ mit $\mu(A) > 0$ und + $g(x) = \infty$ für $x \in A$, dann folgt + $g_k(x) \not\to g(x)$, da $g_k(x) \le S < \infty$ $\contr$). + Also OE $\text{ess sup } g \le S$. + \item Es ist $g_k \xrightarrow{k \to \infty} g$ $\mu$ fast überall, d.h. + \[ + \int_{X}^{} \lim_{k \to \infty} |g_k - g| \d{\mu} = 0 + .\] + \item Betrachte $h \coloneqq 2 S |f|$. Dann gilt $\forall k \in \N$: + \[ + |f| |g_k - g| \le |f| (|g_k| + |g|) \stackrel{\text{(ii)}}{\le} 2 S |f| = h + .\] $h$ ist außerdem integrabel, da $f$ integrabel, also $f_+$ und $f_-$ integrabel, + insbesondere $|f| = f_+ + f_-$ integrabel. + Damit folgt mit dominierter Konvergenz + \begin{salign*} + \lim_{k \to \infty} \int_{X}^{} |f| |g_k - g| \d{\mu} + &\stackrel{\text{3.19}}{=} \int_{X}^{} \lim_{k \to \infty} |f| |g_k - g| \d{\mu} \\ + &\stackrel{\text{(i)}}{\le } \int_{X}^{} M \lim_{k \to \infty} |g_k - g| \d{\mu} \\ + &\stackrel{\text{(iii)}}{=} 0 + .\end{salign*} + \item Da $f_k \to f$ in $L^{1}$ folgt + $\int_{X}^{} |f_k - f| \d{\mu} = \Vert f_k - f \Vert_{L^{1}} \xrightarrow{k \to \infty} 0$. + Damit folgt + \begin{salign*} + \Vert f_k g_k - fg \Vert_{L^{1}} &= \int_{X}^{} |f_k g_k - fg| \d{\mu} \\ + &= \int_{X}^{} |f_k g_k - g_k f + g_k f - fg| \d{\mu} \\ + &\le \int_{X}^{} |g_k| |f_k - f| \d{\mu} + + \int_{X}^{} |f| |g_k - g| \d{\mu} \\ + &\stackrel{\text{(ii)}}{\le} M + \underbrace{\int_{X}^{} |f_k - f | \d{\mu} }_{\xrightarrow{k \to \infty} 0} + + \underbrace{\int_{X}^{} |f| |g_k - g| \d{\mu} }_{\xrightarrow{k \to \infty} 0}\\ + &\xrightarrow{k \to \infty} 0 + .\end{salign*} + Also $f_k g_k \to fg$ in $L^{1}$. + \end{enumerate} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $L^{1}(X, \mu) \cap L^{\infty}(X, \mu) \subseteq L^{p}(X, \mu)$ für + $1 < p < \infty$ und + \[ + \Vert f \Vert_{L^{p}} \le \Vert f \Vert_{L^{1}}^{\frac{1}{p}} + \Vert f \Vert_{L^{\infty}} ^{\frac{p-1}{p}} \qquad + \text{für alle } f \in L^{1}(X, \mu) \cap L^{\infty}(X, \mu) + .\] + \begin{proof} + Sei $1 < p < \infty$ und $f \in L^{1}(X, \mu) \cap L^{\infty}(X, \mu)$. Dann + ist + \begin{salign*} + \int_{X}^{} |f|^{p} \d{\mu} &= + \int_{X}^{} |f|^{1} |f|^{p-1} \d{\mu} \\ + &\stackrel{\text{Hölder}}{\le } \Vert f \Vert_{1} \Vert f^{p-1} \Vert_{\infty} \\ + &= \Vert f \Vert_1 \text{ess sup}_{X} |f^{p-1}| \\ + &\stackrel{p-1>0}{=} \Vert f \Vert_1 \left( \text{ess sup}_X |f| \right)^{p-1} \\ + &= \Vert f \Vert_1 \Vert f \Vert_{\infty}^{p-1} \\ + &\stackrel{p < \infty}{<} \infty + \intertext{Da beide Seiten nicht-negativ, folgt durch Potenzieren mit $\frac{1}{p}$:} + \Vert f \Vert_p &\le \Vert f \Vert_1^{\frac{1}{p}} \Vert f \Vert_{\infty}^{\frac{p-1}{p}} < \infty + .\end{salign*} + Das zeigt beide Behauptungen. + \end{proof} + \item Sei $\mu(X) < \infty$ und $1 \le p \le q \le \infty$. Beh.: + $L^{q}(X, \mu) \subseteq L^{p}(X, \mu)$ und $\exists C \in \R$, s.d. $\forall f \in L^{q}(X, \mu)$ + \[ + \Vert f \Vert_p \le C \Vert f \Vert_q + .\] + \begin{proof} + Sei $f \in L^{q}(X, \mu)$. Dann wähle $r \coloneqq \frac{q}{p} \ge 1$ und + $r' = \frac{q}{q-p}$. Damit folgt + \[ + \frac{1}{r} + \frac{1}{r'} = \frac{p}{q} + \frac{q-p}{q} = 1 + .\] Dann betrachte + \begin{salign*} + \int_{X}^{} |f|^{p} \d{\mu} &= \int_{X}^{} |f|^{p}\cdot 1 \d{\mu} \\ + &\stackrel{\text{Hölder}}{\le } \Vert 1 \Vert_{r'} \Vert f^{p} \Vert_r \\ + &= \Vert 1 \Vert_{r'} \left( \int_{X}^{} (|f|^{p})^{r} \d{\mu} \right)^{\frac{1}{r}} \\ + &\stackrel{r = q / p}{=} + \Vert 1 \Vert_{r'} \left( \int_{X}^{} |f|^{q} \d{\mu} \right)^{\frac{p}{q}} \\ + \intertext{Da beide Seiten nichtnegativ, folgt durch Potenzieren mit $\frac{1}{p}$} + \Vert f \Vert_p &\le \Vert 1 \Vert_{r'}^{\frac{1}{p}} \Vert f \Vert_q + \intertext{Für die Konstante folgt} + \Vert 1 \Vert_{r'}^{\frac{1}{p}} + &= \left( \int_{X}^{} 1^{r'} \d{\mu} \right)^{\frac{1}{pr'}} \\ + &\stackrel{r' = q / (q-p)}{=} \left( \int_{X}^{} 1 \d{\mu} \right)^{\frac{q-p}{pq}} \\ + &= \mu(X)^{\frac{q-p}{pq}} \eqqcolon C < \infty + \intertext{Damit folgt insgesamt} + \Vert f \Vert_p &\le C \Vert f \Vert_q < \infty + .\end{salign*} + Also insbesondere $f \in L^{p}(X, \mu)$. + \end{proof} + Beh.: Es ist $L^{4}((0,1), \lambda) \subsetneqq L^{1}((0,1), \lambda)$. + \begin{proof} + Betrachte $f = x^{-\frac{1}{2}}$. Dann ist + \begin{salign*} + \int_{0}^{1} |f|^{4} \d{\lambda} &= \int_{0}^{1} x^{-2} \d{\lambda} = + - x^{-1}\Big|_{0}^{1} = -1 + \lim_{k \to \infty} \frac{1}{\frac{1}{k}} = \infty \\ + \int_{0}^{1} |f| \d{\lambda} &= \int_{0}^{1} x^{-\frac{1}{2}} \d{\mu} + = 2 (\sqrt{1} - \sqrt{0}) = 2 < \infty + .\end{salign*} + Also ist $f \in L^{1}((0, 1), \lambda)$, aber $f \not\in L^{4}((0,1), \lambda)$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $f_k \to f$ in $L^{1}(X, \mu)$ $\iff$ + \[ + \int_{X}^{} |f_k| \d{\mu} \xrightarrow{ k \to \infty} \int_{X}^{} |f| \d{\mu} + .\] + \begin{proof} + \begin{itemize} + \item ,,$\implies$''. Sei $\Vert f_k - f \Vert_{L^{1}} \xrightarrow{k \to \infty} 0$. Dann + folgt + \begin{salign*} + 0 &\le \left| \int_{X}^{} |f_k| \d{\mu} - \int_{X}^{} |f| \d{\mu} \right| \\ + &= \left| \int_{X}^{} (|f_k| - |f|) \d{\mu} \right| \\ + &\stackrel{\triangle}{\le } \int_{X}^{} | |f_k| - |f| | \d{\mu} \\ + &\stackrel{\triangledown}{=} \int_{X}^{} |f_k - f| \d{\mu} + \xrightarrow{k \to \infty} 0 + .\end{salign*} + \item ,,$\impliedby$''. Sei $\lim_{k \to \infty} \int_{X}^{} |f_k| \d{\mu} + = \int_{X}^{} |f| \d{\mu} $ $(*)$. + Es ist zunächst $|f_k - f| = |f_k + (-f)| \le |f_k| + |f|$. Dann + betrachte $g_k \coloneqq |f_k| + |f| - |f_k -f| \ge 0$. Da + $f_k \xrightarrow{ k \to \infty} f$ $\mu$-f.ü., folgt + $g_k \xrightarrow{k \to \infty} 2 |f|$ $\mu$ f.ü. Damit folgt + mit Lemma von Fatou + \begin{salign*} + \int_{X}^{} 2 |f| \d{\mu} &= \int_{X}^{} \liminf_{k \to \infty} g_k \d{\mu} \\ + &\stackrel{\text{3.18}}{\le } \liminf_{k \to \infty} \int_{X}^{} g_k \d{\mu} \\ + &= \liminf_{k \to \infty} \left( \int_{X}^{} 2|f_k| \d{\mu} + + \int_{X}^{} 2 |f| \d{\mu} - \int_{X}^{} |f_k - f| \d{\mu} \right) \\ + &\stackrel{(*)}{=} \int_{X}^{} |f| \d{\mu} + \int_{X}^{} |f| \d{\mu} + - \limsup_{k \to \infty} \int_{X}^{} |f_k -f| \d{\mu} + \intertext{Umstellen liefert} + 0 &\le \limsup_{k \to \infty} \int_{X}^{} |f_k -f| \d{\mu} \le 0 + \intertext{Wegen $0 \le \liminf_{k \to \infty} \int_{X}^{} |f_k - f|\d{\mu} + \le \limsup_{k \to \infty} \int_{X}^{} |f_k -f| \d{\mu} \le 0$ folgt} + 0 &= \lim_{k \to \infty} \int_{X}^{} |f_k -f| \d{\mu} + .\end{salign*} + Also $f_k \to f$ in $L^{1}(X, \mu)$. + \end{itemize} + \end{proof} + \item Beh.: $f_k \to f$ in $L^{1}(X, \mu)$. + \begin{proof} + Definiere $g_k := g \coloneqq 1$. Dann ist $g = g_k$ integrabel, da + \[ + \int_{X}^{} g \d{\mu} = \int_{X}^{} 1 \d{\mu} = \mu(X) < \infty + .\] Außerdem sind $f_k, f$ integrierbar, insbesondere messbar. Außerdem + gilt $f_k g_k = f_k \cdot 1 = f_k$ und $f g = f \cdot 1 = g$. Dann folgt + die Aussage aus Aufgabe 1. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Beh.: $f \coloneqq \prod_{j=1}^{n} f_j \in L^{p}(X, \mu) $ und + \[ + \Vert f \Vert_{L^{P}(X, \mu)} \le \prod_{j=1}^{n} \Vert f_j \Vert_{L^{p_j}(X, \mu)} + .\] + \begin{proof} + per Induktion über $n$. $n=1$: Dann gilt $p_1 = p$ und $f = f_1$, also trivial. + Sei nun $n \in \N$ mit Aussage gezeigt für $n$. Sei $1 \le p \le \infty$ und seien $1 \le p_j < \infty$ + für $j \in \{ 1, \ldots, n+1\} $ und + \[ + \sum_{j=1}^{n+1} \frac{1}{p_j} = \frac{1}{p} + .\] Dann ist $p \le p_j$ für $j \in \{1, \ldots, n+1\} $. + Definiere $q \coloneqq \frac{p p_{n+1}}{p_{n+1}-p} \ge 1$. Dann ist + \begin{salign*} + \frac{1}{q} = \frac{p_{n+1} - p}{p p_{n+1}} = \frac{1}{p} - \frac{1}{p_{n+1}} = + \sum_{j=1}^{n} \frac{1}{p_j} + .\end{salign*} + Außerdem definiere $r \coloneqq \frac{p_{n+1}}{p} \ge 1$. Dann ist + $r' = \frac{p_{n+1}}{p_{n+1}-p}$, denn + $\frac{1}{r} + \frac{1}{r'} = \frac{p_{n+1} -p}{p_{n+1}} + \frac{p}{p_{n+1}} = 1$. + Seien nun $f_j \in L^{p_j}(X, \mu)$ für $j \in \{1, \ldots, n+1\} $. + Damit betrachte + \begin{salign*} + \int_{X}^{} |f|^{p} \d{\mu} &= \int_{X}^{} \left| \prod_{j=1}^{n+1} f_j \cdot f_{n+1}\right| \d{\mu} \\ + &\stackrel{\text{Hölder}}{\le } \left\Vert \left( \prod_{j=1}^{n} f_j \right)^{p} \right\Vert_{r'} + \left\Vert f_{n+1}^{p} \right\Vert_r \\ + &= \left( \int_{X}^{} \left| \prod_{j=1}^{n} f_j \right|^{\frac{p p_{n+1}}{p_{n+1}-p}} \d{\mu} \right)^{\frac{p_{n+1} -p}{p_{n+1}}} \left( \int_{X}^{} |f_{n+1}|^{p_{n+1}} \d{\mu} \right)^{\frac{p}{p_{n+1}}} \\ + \intertext{Da beide Seiten nicht-negativ sind, folgt durch Potenzieren mit $\frac{1}{p}$} + \Vert f \Vert_p &\le \left\Vert \prod_{j=1}^{n} f_j \right\Vert_q \Vert f_{n+1} \Vert_{p_{n+1}} \\ + &\stackrel{\text{IV}}{=} \prod_{j=1}^{n} \Vert f_j \Vert_{p_j} \Vert f_{n+1} \cdot \Vert_{p_{n+1}} \\ + &= \prod_{j=1}^{n+1} \Vert f_j \Vert_{p_j} + .\end{salign*} + \end{proof} +\end{aufgabe} + +\end{document} diff --git a/ws2020/tut-ana/loesung5.xopp b/ws2020/tut-ana/loesung5.xopp new file mode 100644 index 0000000000000000000000000000000000000000..5d409cf026ea7ff16268047c20faeaa466034622 GIT binary patch literal 189635 zcmV(-K-|9{iwFP!000001GK&CQY*WbF8IHmV#ey9^qJ1oHPx|qov<-B1_K6TZhubr ze!!g>2mSU zKjPofAHUX{?qblLFBijq{;^p8@jw6bKmKu8Za)^?`LF)Ru)A4){PT||_lJ{{U-57H z(#c@B`{UPWIv*Y%CH|oAZKj*~@IUhp50?Gy+i7nf3lx}(_FVWYeGCs7H(Xv{a54x9TZsyCgRpa!v^uXUf&Gk6> zd|E%nPiMpJ#>n1G)i%3i@?4K(e}CQG?&Dg!wiyoY)L8KA$<4MLcgm}zvgo|sw#rRW z-ac2pmSyu^=|pATsn69-{*5$gWzKk2g zkL31HFI^mVYW#e|^sGBR<1t?{uIr1-@;EBbu0EOAvhs9v@>yE&O=q^|cgy|r z)$qAIA9dBfHC!KdL(S%*H@ZsNlWMDXx+#72H>KU#Ci%Ku2;J@9Jm_Kh`8h6k_lx1^ zz0JQS@6!jqze`T<$JLLs)Bdpf^v=#%g}`U^F5cH#ch>#`EuN=m2XBO{y3aI-#%{M>*``W zi5tmBul!IyYkrvK?e6~WVl^+ThkgAmdU;Kz&HMWE$GGvlk58}X@BPch)l=*8@oIi` 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