Lecture12

# Lecture12 - \documentclass[12pt,letterpaper]cfw_article

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\alpha_2 : X_2 & \ra X_1\times X_2 : h_2 \ramp (0,h_2). \end{align*} $\alpha_1$ and $\alpha_2$ are linear and \begin{align*} \left\| \alpha_1(h_1) \right\| & = \left\| (h_1,0) \right\| = \left\| h_1 \right\| \\ \left\| \alpha_2(h_2) \right\| & = \left\| (0,h_2) \right\| = \left\| h_2 \right\| . \end{align*} It follows that $\left\| \alpha_1 \right\| = 1 = \left\| \alpha_2 \right\|$. We can see that \begin{align*} Since $F'$ is continuous at each $a$ let $\varepsilon > 0$ and choose $\delta$ such that if $b \in U$ with $||a - b|| < \delta$ then $\left\| F'_a - F'_b \right\| < \varepsilon,$ thus for $i = 1$ or $2$ \begin{align*} \left\| \partial_i F_a - \partial_i F_b \right\| & = \left\| F'_a \circ \alpha_i - F'_b \circ \alpha_i \right\| \\ & = \left\| \left( F'_a - F'_b \right) \circ \alpha_i \right\| \\ & \le \left\| F'_a - F'_b \right\| \left\| \alpha_i \right\| \\ & = \left\| F'_a - F'_b \right\|
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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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Lecture12 - \documentclass[12pt,letterpaper]cfw_article

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