Lecture06

# 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 This preview has intentionally blurred sections. Sign up to view the full version. View Full Document 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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