Lecture04

# Lecture04 - \documentclass[12pt,letterpaper]cfw_article\usepackagecfw_amsmath\usepackagecfw_amssymb\usepackagecfw_latexsym\usepackagecfw_array\usepa

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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 4} \author{} \maketitle \ \begin{definition} \ Let $a$ be a point in the open subset $U$ of the normed space $X$ and let $Y$ be a normed space. The function $F: U \ra Y$ is said to be {\bf differentiable at a} iff there exists a bounded linear map $A \in L(X,Y)$ such that t $\lim_{\| h \| \rightarrow 0} \frac{F(a+h) - F(a) - A(h)}{\| h \| } = 0.$ \ \end{definition} \ More explicitly, this says that for each $\varepsilon > 0$, there exists $\delta > 0$ such that 0 $\| h \| \leq \delta \Ra \| F(a+h) - F(a) - A(h) \| \leq \varepsilon \| h \|$. We call $A$ the derivative of $F$ at $a$ and denote it as $F_{a}'$. W \medbreak \noindent {\bf Exercise}: If $A$ exists, then it is unique. I \begin{proof} Let $A_{1}$ and $A_{2}$ be derivatives for $F:U \ra Y$ at $a$. Then, given $\varepsilon > 0$ $\|A_{1}(h) - A_{2}(h) \| = \| (F(a+h) - F(a) - A_{2}) - (F(a+h) - F(a) - A_{1}) \|$

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$\leq \| F(a+h) - F(a) - A_{2} \| + \| F(a+h) - F(a) - A_{1} \| \leq 2h \varepsilon$ \ \noindent provided that $\| h \| \leq \delta$ for some $\delta > 0$. This implies that $\| A_{1} - A_{2} \| = 0$ so that $A_{1} = A_{2}$ and the derivative of $F$ at $a$ is
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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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Lecture04 - \documentclass[12pt,letterpaper]cfw_article\usepackagecfw_amsmath\usepackagecfw_amssymb\usepackagecfw_latexsym\usepackagecfw_array\usepa

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