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'operator norm'
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| Title of object: |
operator norm |
| Canonical Name: |
OperatorNorm |
| Type: |
Definition |
| Created on: |
2002-06-03 20:12:52-04 |
| Modified on: |
2002-06-03 20:12:52-04 |
| Classification: |
msc:47L25 |
| Defines: |
bounded linear map |
| Synonyms: |
operator norm=induced norm |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\def\V{{\mathsf V}}
\def\W{{\mathsf W}}
\def\R{{\mathbb R}}
\def\v{{\mathbf v}}
\def\op{{\rm op}} |
Content:
Let $A\colon \V\to\W$ be a linear map between normed vector spaces $\V$ and
$\W$. We can define a function $\|\cdot\|_{\op}\colon A\mapsto \R$ as
\|A\|_{\op}=\sup_{\v\in\V} \frac{\|A\v\|}{\|\v\|}.
Turns out that $\|\cdot\|_{\op}$ satisfies all the properties of a norm and hence
is called the o{\em perator norm} (or the {\em induced norm}) of $A$. If
$\|A\|_{op}$ exists and is finite, we say that $A$ is a {\em bounded linear map}.
The space $L(\V,\W)$ of bounded linear maps from $\V$ to $\W$ also forms
a vector space with $\|\cdot\|_{\op}$ as the natural norm.
\subsection{Example 1}
Suppose that $\V=(\R^n,\|\cdot\|_p)$ and $\W=(\R^n,\|\cdot\|_p)$, where
$\|\cdot\|_p$ is the vector p-norm. Then the operator norm
$\|\cdot\|_\op = \|\cdot\|_p$ is the matrix p-norm. |
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