Lecture06 - \documentclass[12pt,letterpaper]cfw_article \

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\documentclass[12pt,letterpaper]{article} \usepackage{amsmath,amssymb,latexsym,array,enumerate,amsthm,amscd} \ \usepackage[all]{xy} \ \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \ \newcommand{\bR}{\mathbb{R}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \ \date{} \author{} \ \title{\bf Lecture 6} \ \begin{document} \ \maketitle \ \noindent {\bf Miscellany} { [ {\bf (1)} If $S:X \times Y \ra Z$ is (separately) continuous, then $S$ is bounded when $X$ is complete. $ \begin{proof} This uses the {\bf Principle of Uniform Boundedness (PUB)}. \ {\bf PUB}: If the collection $\{A_\lambda : \lambda\in \Lambda \}\subset L(X,Z)$ is pointwise bounded in the sense that if $x\in X$ then the set \[\{A_{\lambda}(x) : \lambda \in \Lambda \}\subset Z \] is bounded (i.e. for all $x\in X$ there is an $M_{x}$ such that \[ \lambda\in\Lambda \Ra \|A_{\lambda}x\|\leq M_{x}), \] then it is uniformly bounded in the sense that there exists $M$ such that \[\lambda \in \Lambda \Ra \|A_{\lambda}\| \leq M \] \underline{provided} $X$ is complete. \ $\newline$ Recall from last time that we had $T_{x}(y)=S(x,y)$ and $T_{x}\in L(Y,Z)$. So we have $T: X \ra L(Y,Z)$. Consider the set \[ \{S(\cdot,y) : y\in Y, \|y\|\leq1 \}\subset L(X,Z). \] This is pointwise bounded: if $x\in X$ is fixed, then \[y\in Y, \|y\|\leq 1 \Ra \|S(\cdot,y)(x)\|=\|S(x,y)\|=\|T_{x}(y)\|\leq \|T_{x}\|. \] So (PUB) there is $M$ such that \[y\in Y, \|y\|\leq 1 \Ra \|S(\cdot,y)\|\leq M. \] Thus T \[y\in Y \Ra \|S(\cdot,y)\|\leq M \|y\|. \] and so \[x\in X, y\in Y \Ra \|S(x,y)\|=\|S(\cdot,y)(x)\|\leq \|S(\cdot,y)\|\|x\|\leq M\|y\|\|x\|. \] \end{proof} \ In the proof above we assumed $X$ is complete to be able to use PUB. However notice that we could just
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as well have assumed $Y$ was complete and changed the roles of $X$ and $Y$. a
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This note was uploaded on 11/09/2011 for the course MAT 6932 taught by Professor Staff during the Spring '10 term at University of Florida.

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Lecture06 - \documentclass[12pt,letterpaper]cfw_article \

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