Lecture14 - \documentclass[12pt,letterpaper]cfw_article

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
\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} \newtheorem*{remark}{Remark:} \newtheorem*{application}{Application:} \newtheorem*{fact}{Fact:} \ \newcommand{\bC}{{\mathbb C}} \newcommand{\bH}{{\mathbb H}} \newcommand{\bR}{{\mathbb R}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \ \date{} \ \begin{document} \ \title{\bf Lecture 14} \author{} \maketitle \ \begin{theorem} Let X be a Banach space. The map \[\exp:L(X) {\ra} L(X): A \mapsto e^A \] is differentiable at 0 and \[exp'_0=id_{L(X)}. \] \end{theorem} \ \begin{proof} We must estimate $ \displaystyle { \frac{\| e^H-e^0-id_{L(X)}H \|}{\| H \| } } $. \newline Because $ \displaystyle {e^H=\sum_{n=0}^{\infty}{\frac{1}{n!}H^n}}$, we have that \[e^H-I-H = \sum_{n=2}^{\infty}{\frac{1}{n!}H^n} = H^2\sum_{n=0}^{\infty}{\frac{1}{(n+2)!}H^n}\] \[ \therefore \| e^H-I-H \| \leq \|H\|^2 \sum_{n=0}^{\infty}{\frac{1}{(n+2)!}\|H\| ^n} \leq \|H\|^2e^{\|H\|}\] \[ \therefore \frac{\| e^H-I-H \|}{\|H\|} \leq \|H\|e^{\|H\|}.\] As $H \ra 0$, $\|H\| \ra 0$ and $e^{\|H\|} \ra 1$. Hence $\frac{\| e^H-I-H \|}{\| H\|} \ra 0$. \end{proof} \ \noindent {\bf Fact}: The function $\exp$ is differentiable everywhere. \
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
\begin{theorem} Let $A \in L(X)$ and define $f(t)=e^{tA}$ for $t \in \bR$. Then \newline $f'(t)=Ae^{tA}$ for all $t \in \bR$. \end{theorem}
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 4

Lecture14 - \documentclass[12pt,letterpaper]cfw_article

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online