Lecture01 - \documentclass[12pt,letterpaper]{article} \...

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Unformatted text preview: \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 1} \author{} \maketitle \ \begin{definition} A {\bf norm} on the real vector space $V$ is a map \[\| \bullet \| : V \ra \bR \] with the following properties (for $t \in \bR$ and $x, y, z \in V$): \[\| z \| \geqslant 0 \; \; and \; \; \| z \| = 0 \Leftrightarrow z = 0 ; \] \[\| t z \| = |t| \|z \| ; \] \[\| x + y \| \leqslant \| x \| + \| y \| . \] \ \end{definition} \ The rule \[x, y \in V \Ra {\rm d} (x, y) = \| x - y \| \] then defines a metric ${\rm d}$ on $V$; this is an easy {\it exercise}. All subsequent metric properties of a normed space refer to this. s \begin{definition} A {\bf Banach space} is a complete normed space. \end{definition} \ This is the first such metric reference; completeness is of course in the Cauchy sense. s \begin{definition} An {\bf inner product} on the real vector space $V$ is a real-bilinear map \[( \bullet | \bullet ): V \times V \ra \bR...
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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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Lecture01 - \documentclass[12pt,letterpaper]{article} \...

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