|
|
|
Viewing Version
5
of
'orthonormal basis'
|
[ view 'orthonormal basis'
|
back to history
]
| Title of object: |
orthonormal basis |
| Canonical Name: |
OrthonormalBasis |
| Type: |
Definition |
| Created on: |
2003-10-15 01:32:21 |
| Modified on: |
2006-11-27 08:09:58 |
| Classification: |
msc:15A03 |
Revision comment (for changes between this and next version):
Preamble:
%fancy typeface/symbols
%\usepackage{beton}
%\usepackage{concmath}
%\usepackage{stmaryrd}
%margins
%\usepackage{simplemargins}
%\usepackage{multicol}
%\setleftmargin{1in}
%\setrightmargin{1in}
%\settopmargin{1in}
%\setbottommargin{1in}
% symbols
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage[mathscr]{euscript}
\usepackage[T1]{fontenc}
% graphics
%\usepackage{pstricks}
% math
\usepackage{amsmath}
\usepackage{amsopn}
\usepackage{amstext}
\usepackage{amsthm}
% only one theoremstyle
%\theoremstyle{definition}
%\newtheorem{exercise}{}[subsection]
% math operators
\DeclareMathOperator{\pipe}{{\big |} \hspace{-2.85pt} {\big |}}
\DeclareMathOperator{\et}{\&}
% misc math stuff
\newcommand{\vol}{\mathrm{vol}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\domain}{\mathrm{domain}}
\newcommand{\supp}{\mathrm{supp}}
\newcommand{\diam}{\mathrm{diameter}}
\newcommand{\incl}{\mathrm{incl}}
\newcommand{\interior}{\mathrm{interior}}
\newcommand{\triv}{\mathrm{triv}}
\newcommand{\image}{\mathrm{image}}
\newcommand{\closure}{\mathrm{closure}}
\newcommand{\degree}{\mathrm{degree}}
\newcommand{\im}{\mathrm{im}}
\newcommand{\esssup}{\mathrm{ess\ sup}}
\newcommand{\openset}{\mathrel{{\mathchoice{\rlap{$\subset$}{\;\circ}}%
{\rlap{$\subset$}{\;\circ}}%
{\rlap{$\scriptstyle\subset$}{\;\circ}}%
{\rlap{$\scriptscriptstyle\subset$}{\;\circ}}}}}
% foreignisms
\newcommand{\ie}{\emph{i.e.},}
\newcommand{\Ie}{\emph{I.e.},}
\newcommand{\cf}{\emph{cf.}}
\newcommand{\eg}{\emph{e.g.},}
\newcommand{\Eg}{\emph{E.g.},}
\newcommand{\ala}{\emph{\'a la}}
% labels for spaces
\newcommand{\N}{\mathbf{N}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\Q}{\mathbf{Q}}
\newcommand{\C}{\mathbf{C}}
\newcommand{\CP}{\mathbf{CP}}
\newcommand{\RP}{\mathbf{RP}}
\newcommand{\R}{\mathbf{R}}
\newcommand{\Rtw}{\mathbf{R}^2}
\newcommand{\Rth}{\mathbf{R}^3}
\newcommand{\Rfo}{\mathbf{R}^4}
\newcommand{\Rfi}{\mathbf{R}^5}
\newcommand{\Rp}{\mathbf{R}^p}
\newcommand{\Rn}{\mathbf{R}^n}
\newcommand{\Rnmon}{\mathbf{R}^{n-1}}
\newcommand{\Rnpon}{\mathbf{R}^{n+1}}
\newcommand{\Rmpon}{\mathbf{R}^{m+1}}
\newcommand{\RNpon}{\mathbf{R}^{N+1}}
\newcommand{\Rtwn}{\mathbf{R}^{2n}}
\newcommand{\RN}{\mathbf{R}^N}
\newcommand{\Rm}{\mathbf{R}^m}
\newcommand{\Hon}{\mathbf{H}^1}
\newcommand{\Htw}{\mathbf{H}^2}
\newcommand{\Hth}{\mathbf{H}^3}
\newcommand{\Hn}{\mathbf{H}^n}
\newcommand{\Torus}{\mathbf{T}}
\newcommand{\Ttw}{\mathbf{T}^2}
\newcommand{\Tth}{\mathbf{T}^3}
\newcommand{\Tn}{\mathbf{T}^n}
\newcommand{\Son}{\mathbf{S}^1}
\newcommand{\Stw}{\mathbf{S}^2}
\newcommand{\Sth}{\mathbf{S}^3}
\newcommand{\Sfo}{\mathbf{S}^4}
\newcommand{\Sfi}{\mathbf{S}^5}
\newcommand{\SN}{\mathbf{S}^N}
\newcommand{\Sn}{\mathbf{S}^n}
\newcommand{\Sm}{\mathbf{S}^m}
\newcommand{\Smmon}{\mathbf{S}^{m-1}}
\newcommand{\Snmon}{\mathbf{S}^{n-1}}
\newcommand{\Snmtw}{\mathbf{S}^{n-2}}
\newcommand{\Hnmon}{\mathbf{H}^{n-1}}
\newcommand{\Ton}{\mathbf{T}^1}
\newcommand{\T}{\mathbf{T}}
% differential operators
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\pkd}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}}
\newcommand{\dkd}[3]{\frac{d^{#3} #1}{d#2^{#3}}}
\newcommand{\td}[2]{\frac{d #1}{d #2}}
\newcommand{\pdat}[3]{\left . \frac{\partial #1}{\partial #2}\right|_{#3}}
\newcommand{\dbyd}[1]{ \frac{d}{d #1}}
% Fri Oct 3 11:00:53 2003 -- should check at some point to see whether the replaced versions are actually used at all, I don't think so
\newcommand{\ddk}[3]{\frac{d^{#3} #1}{d#2^{#3}}}
% replaces...
\newcommand{\dbydk}[2]{ \frac{d^{#2}}{d #1^{#2}}}
\newcommand{\ppk}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}}
% replaces...
\newcommand{\pbypk}[2]{ \frac{\partial^{#2}}{\partial #1^{#2}}}
\newcommand{\pp}[2]{\frac{\partial #1}{\partial #2}}
% replaces...
\newcommand{\pbyp}[1]{ \frac{\partial}{\partial #1}}
\newcommand{\ddat}[3]{\left .\frac{d #1}{d #2}\right|_{#3}}
% replaces...
\newcommand{\dbydat}[2]{\left . \frac{d}{d #1} \right|_{#2}}
\newcommand{\ppkat}[4]{\left .\frac{\partial^{#3} #1}{\partial #2^{#3}}\right|_{#4}}
\newcommand{\ddkat}[4]{\left .\frac{d^{#3} #1}{d#2^{#3}}\right|_{#4}}
% counters for new lists
%\newcounter{alistctr}
%\newcounter{Alistctr}
\newcounter{rlistctr}
\newcounter{Rlistctr}
\newcounter{123listctr}
\newcounter{123listcolonstylectr}
% a,b,c - small latin letter list
\newenvironment{alist}{
\indent \begin{list}{(\alph{alistctr})}{\usecounter{alistctr}}}
{\end{list}\setcounter{alistctr}{0}}
% A,B,C - LARGE LATIN LETTER LIST
\newenvironment{Alist}{
\indent \begin{list}{(\Alph{Alistctr})}{\usecounter{Alistctr}}
}
{\end{list}\setcounter{Alistctr}{0}}
% i,ii,iii - small roman numeral list
\newenvironment{rlist}{
\indent \begin{list}{(\roman{rlistctr})}{\usecounter{rlistctr}}
}
{\end{list}\setcounter{rlistctr}{0}}
% I,II,III - large roman numeral list
\newenvironment{Rlist}{
\indent \begin{list}{(\Roman{Rlistctr})}{\usecounter{Rlistctr}}
}
{\end{list}\setcounter{Rlistctr}{0}}
%1,2,3 - arabic numeral list
\newenvironment{123list}{
\indent \begin{list}{(\arabic{123listctr})}{\usecounter{123listctr}}
}
{\end{list}\setcounter{123listctr}{0}}
%1:,2:,3: - arabic numeral list with colon decoration
\newenvironment{123listcolonstyle}{\indent \begin{list}{\arabic{123listcolonstylectr}:}{\usecounter{123listcolonstylectr}}}
{\end{list}\setcounter{123listcolonstylectr}{0}}
% environment for definitions
\def\defn#1{\addcontentsline{toc}{subsection}{$\ast$} {\footnotesize \noindent \begin{123listcolonstyle} \setlength{\itemsep}{0em} \setlength{\topsep}{0em} \setlength{\parsep}{0em} #1 \end{123listcolonstyle}}}
%environment for proofs
\def\proof#1{\par {\footnotesize \indent \begin{tabular}{ll} #1 \end{tabular}}}
%% \include{packages}
%% \include{renewed_commands}
%% \include{simple_theorems}
%% \include{abbreviations}
%% \include{spaces}
%% \include{new_environments}
%% \include{margins}
%% \include{mathoperators}
%% \include{differentiation}
%% \include{limited_things}
%% \include{argawarga} |
Content:
\section*{orthonormal basis}
Let $X$ be an inner product space over a field $F$ and
\[\{x_{\alpha}\}_{\alpha\in J} \subset X\] be a set of orthonormal vectors in the space. If we can write any vector in our space as the sum of vectors from the set multiplied by elements of the field, or in symbols
\[
\forall x\in X:\exists \{a_{\alpha}\}_{\alpha\in J}\subset F:x=\sum_{\alpha\in J} a_{\alpha} x_{\alpha}
\] then we say that $\{x_{\alpha}\}$ form an orthonormal basis for $X$. |
|
|
|
|
|
