Lecture01

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

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full 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 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online