Lecture09

# Lecture09 - \documentclass[12pt,letterpaper]cfw_article

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\documentclass[12pt,letterpaper]{article} \ \usepackage{amsmath} \usepackage{amssymb} \usepackage{latexsym} \usepackage{array} \usepackage{enumerate} \usepackage{amsthm} \usepackage{amscd} \ \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \ \newcommand{\bC}{{\mathbb C}} \newcommand{\bH}{{\mathbb H}} \newcommand{\bR}{{\mathbb R}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \ \date{} \ \begin{document} \ \title{\bf Lecture 9} \author{} \maketitle \ Recall: $\left\{ \begin{array}{l} L=[a,b] \subset U \stackrel{F}{\rightarrow} Y \\ F \textrm{ differentiable at each point of } L\\ \Psi \in Y^{*} \end{array} \right.$ \ $\Ra \exists p \in L \textrm{ such that } \Psi(F(b)-F(a)) = \Psi(F_{p}'(b-a))$ \ \begin{theorem}[MVT] If $F:U \ra Y$ is differentiable at each point of $[a,b] \subset U$ then there exists $p \in (a,b)$ such that $\|F(b)-F(a) \|\leq \|F_{p}'(b-a) \| \quad (\leq \|F_{p}'\| \|b-a \|)$ \end{theorem} \ \begin{proof} Let $\Psi \in Y^{*}$ be such that $\| \Psi \| =1$ and $\Psi(F(b)-F(a)) = \|F(b) - F(a)\|.$ Apply previous MVT: there exists $p \in (a,b)$ such that \begin{eqnarray*} \end{eqnarray*}

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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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Lecture09 - \documentclass[12pt,letterpaper]cfw_article

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