Lecture07 - \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 7} \author{} \maketitle \ \underline{Interlude}: Uniqueness of the derivative.\\ There is at most one $A\in L(X,Y)$ such that\\ $$\dfrac{F(a+h)-F(a)-A(h)}{\|h\|}\ra 0$$ as $h\ra 0$ a \begin{proof}[Proof 1] Lecture 4 notes \end{proof} \begin{proof}[Proof 2] Suppose $A:X\ra Y$ is linear and satisfies \[\lim_{h\ra 0} \frac{F(a+h)-F(a)-A(h)}{\|h\|}=0. \] Fix $a,h$. Introduce $t\in \bR$. Then as $t\ra 0$, \[\tfrac{1}{\|th\|}( F(a+th)-F(a)-A(th))\ra 0 \] and \[\tfrac{1}{\|th\|}\| F(a+th)-F(a)-A(th)\|=\tfrac{1}{|t|\|h\|}\| F(a+th)-F(a)- tA(h)\| \] \[=\frac{1}{\|h\|} \Big\| \frac {F(a+th)-F(a)}{t}-A(h) \Bigr\| \] so \[A(h)=\lim_{t\ra 0} \frac{F(a+th)-F(a)}{t}. \] So $F_{a}^{'}:X\ra Y$ is given by
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Lecture07 - \documentclass[12pt,letterpaper]cfw_article

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