\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Hilbert space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{hilbert_spaces_as_banach_spaces}{Hilbert spaces as Banach spaces}\dotfill \pageref*{hilbert_spaces_as_banach_spaces} \linebreak
\noindent\hyperlink{hilbert_spaces_as_metric_spaces}{Hilbert spaces as metric spaces}\dotfill \pageref*{hilbert_spaces_as_metric_spaces} \linebreak
\noindent\hyperlink{hilbert_spaces_as_conformal_spaces}{Hilbert spaces as conformal spaces}\dotfill \pageref*{hilbert_spaces_as_conformal_spaces} \linebreak
\noindent\hyperlink{morphisms_of_hilbert_spaces}{Morphisms of Hilbert spaces}\dotfill \pageref*{morphisms_of_hilbert_spaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{banach_spaces}{Banach spaces}\dotfill \pageref*{banach_spaces} \linebreak
\noindent\hyperlink{of_lebesgue_squareintegrable_functions_over_a_manifold}{Of Lebesgue square-integrable functions over a manifold}\dotfill \pageref*{of_lebesgue_squareintegrable_functions_over_a_manifold} \linebreak
\noindent\hyperlink{of_squareintegrable_halfdensities}{Of square-integrable half-densities}\dotfill \pageref*{of_squareintegrable_halfdensities} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{bases}{Bases}\dotfill \pageref*{bases} \linebreak
\noindent\hyperlink{cauchyschwarz_inequality}{Cauchy--Schwarz inequality}\dotfill \pageref*{cauchyschwarz_inequality} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\begin{quote}%
\emph{Dr. von Neumann, ich m\"o{}chte gerne wissen, was ist denn eigentlich ein Hilbertscher Raum ?} \footnote{\emph{Dr. von Neumann, I would like to know what is a Hilbert space ?} Question asked by Hilbert in a 1929 talk by v. Neumann in G\"o{}ttingen. The anecdote is narrated together with additional information on the introduction of adjoint operators to quantum mechanics by Saunders Mac Lane in \emph{Concepts and Categories} (\href{http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf}{link}, p.330). Note, that we have corrected `dann' in the original quotation to the more likely `denn'.} 


\end{quote}
A \emph{Hilbert space} is a (possibly) infinite-dimensional generalisation of the traditional spaces of Euclidean geometry in which the notions of distance and angle still make good sense. This is done through an algebraic operation, the \emph{inner product}, that generalises the dot product.

Hilbert spaces were made famous to the world at large through their applications to [[physics]], where they organise the pure states of quantum systems.

See also

\begin{itemize}%
\item [[an elementary treatment of Hilbert spaces]].

\end{itemize}
\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $V$ be a [[vector space]] over the field of [[complex number]]s. (One can generalise the choice of [[field]] somewhat.) An \textbf{inner product} (in the most general, possibly indefinite, sense) on $V$ is a function

\begin{displaymath}
\langle {-},{-} \rangle: V \times V \to \mathbb{C}
\end{displaymath}
that is (1--3) \emph{sesquilinear} and (4) \emph{conjugate-symmetric}; that is:

\begin{enumerate}%
\item $\langle 0, x \rangle = 0$ and $\langle x, 0 \rangle = 0$;
\item $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$ and $\langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle$;
\item $\langle c x, y \rangle = \bar{c} \langle x, y \rangle$ and $\langle x, c y \rangle = c \langle x, y \rangle$;
\item $\langle x, y \rangle = \overline{\langle y, x \rangle}$.

\end{enumerate}
Here we use the \emph{physicist's convention} that the inner product is conjugate-linear in the first variable rather than in the second, rather than the \emph{mathematician's convention}, which is the reverse. The physicist's convention fits in a little better with $2$-[[2-Hilbert space|Hilbert space]]s. Note that we use the same field as values of the inner product as for [[scalars]]; the complex conjugation will be irrelevant for some choices of field.

The axiom list above is rather redundant. First of all, (1) follows from (3) by setting $c = 0$; besides that, (1--3) come in pairs, only one of which is needed, since each half follows from the other using (4). It is even possible to derive (3) from (2) by supposing that $V$ is a [[topological vector space]] and that the inner product is continuous (which, as we will see, is always true anyway for a Hilbert space).

The next concept to define is (semi)definiteness. We define a function $\|{-}\|^2: V \to \mathbb{C}$ by $\|x\|^2 = \langle x, x \rangle$; in fact, $\|{-}\|^2$ takes only real values, by (4). * The inner product is \textbf{positive semidefinite}, or simply \textbf{positive}, if $\|x\|^2 \geq 0$ always. * Notice that (by 1), $\|x\|^2 = 0$ if $x = 0$; the inner product is \textbf{definite} if the converse holds. * An inner product is \textbf{positive definite} if it is both positive and definite. * As an aside, there are also \emph{negative (semi)definite} inner products, which are slightly less convenient but not really different. An inner product is \emph{indefinite} if some $\|x\|^2$ are positive and some are negative; these have a very different flavour.

The inner product is \textbf{complete} if, given any infinite [[sequence]] $(v_1, v_2, \ldots)$ such that

\begin{equation}
\lim_{m,n\to\infty} \left\|\sum_{i=m}^{m+n} v_i\right\|^2 = 0 ,
\label{Cauchy}\end{equation}
there exists a (necessarily unique) \textbf{sum} $S$ such that

\begin{equation}
\lim_{n\to\infty} \left\|S - \sum_{i=1}^n v_i\right\|^2 = 0 .
\label{converge}\end{equation}
If the inner product is definite, then this sum, if it exists, must be unique, and we write

\begin{displaymath}
S = \sum_{i=1}^\infty v_i
\end{displaymath}
(with the right-hand side undefined if no such sum exists).

Then a \textbf{Hilbert space} is simply a vector space equipped with a complete positive definite inner product.

\hypertarget{hilbert_spaces_as_banach_spaces}{}\subsubsection*{{Hilbert spaces as Banach spaces}}\label{hilbert_spaces_as_banach_spaces}

If an inner product is positive, then we can take the principal square root of $\|x\|^2 = \langle x, x \rangle$ to get the a real number $\|x\|$, the \textbf{norm} of $x$.

This norm satisfies all of the requirements of a [[Banach space]]. It additionally satisfies the \emph{parallelogram law}

\begin{displaymath}
\|x + y\|^2 + \|x - y\|^2 = 2 \|x\|^2 + 2 \|y\|^2 ,
\end{displaymath}
which not all Banach spaces need satisfy. (The name of this law comes from its geometric interpretation: the norms in the left-hand side are the lengths of the diagonals of a parallelogram, while the norms in the right-hand side are the lengths of the sides.)

Furthermore, any Banach space satsifying the parallelogram law has a unique inner product that reproduces the norm, defined by

\begin{displaymath}
\langle x, y \rangle = \frac{1}{4}\left(\|x + y\|^2 - \|x - y\|^2 - \mathrm{i} \|x + \mathrm{i}y\|^2 + \mathrm{i} \|x - \mathrm{i}y\|^2\right) ,
\end{displaymath}
or $\frac{1}{2}(\|x + y\|^2 - \|x - y\|^2)$ in the real case.

Therefore, it is possible to \emph{define} a Hilbert space as a Banach space that satisfies the parallelogram law. This actually works a bit more generally; a positive semidefinite inner product space is a pseudonormed vector space that satisfies the parallelogram law. (We cannot, however, recover an indefinite inner product from a norm.)

\hypertarget{hilbert_spaces_as_metric_spaces}{}\subsubsection*{{Hilbert spaces as metric spaces}}\label{hilbert_spaces_as_metric_spaces}

In any positive semidefinite inner product space, let the \textbf{distance} $d(x,y)$ be

\begin{displaymath}
d(x,y) = \|y - x\| .
\end{displaymath}
Then $d$ is a [[pseudometric]]; it is a complete metric if and only if we have a Hilbert space.

In fact, the axioms of a [[Banach space]] (or pseudonormed vector space) can be written entirely in terms of the metric; we can also state the parallelogram law as follows:

\begin{displaymath}
d(x,y)^2 + d(x,-y)^2 = 2 d(x,0)^2 + 2 d(x,x+y)^2 .
\end{displaymath}
In definitions, it is probably most common to see the metric introduced only to state the completeness requirement. Indeed, \eqref{Cauchy} says that the sequence of partial sums is a [[Cauchy sequence]], while \eqref{converge} says that the sequence of partial sums converges to $S$.

\hypertarget{hilbert_spaces_as_conformal_spaces}{}\subsubsection*{{Hilbert spaces as conformal spaces}}\label{hilbert_spaces_as_conformal_spaces}

Given two vectors $x$ and $y$, both nonzero, let the \textbf{angle} between them be the angle $\theta(x,y)$ whose cosine is

\begin{displaymath}
\cos \theta(x,y) = \frac { \langle x, y \rangle } { \|x\| \|y\| } .
\end{displaymath}
(Note that this angle may be imaginary in general, but not for a Hilbert space over $\mathbb{R}$.)

A Hilbert space cannot be reconstructed entirely from its angles, however (even given the underlying vector space). The inner product can only be recovered up to a positive scale factor.

\hypertarget{morphisms_of_hilbert_spaces}{}\subsubsection*{{Morphisms of Hilbert spaces}}\label{morphisms_of_hilbert_spaces}

See discussion at [[Banach space]]. There is more to be said here concerning duals (including why the theory of Hilbert spaces is slightly nicer over $\mathbb{C}$ while that of Banach spaces is slightly nicer over $\mathbb{R}$).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{banach_spaces}{}\subsubsection*{{Banach spaces}}\label{banach_spaces}

All of the $p$-parametrised examples at [[Banach space]] apply if you take $p = 2$.

In particular, the $n$-dimensional [[vector space]] $\mathbb{C}^n$ is a complex Hilbert space with

\begin{displaymath}
\langle x, y \rangle = \sum_{u=1}^n \bar{x}_u y_u .
\end{displaymath}
Any subfield $K$ of $\mathbb{C}$ gives a positive definite inner product space $K^n$ whose completion is either $\mathbb{R}^n$ or $\mathbb{C}^n$. In particular, the [[cartesian space]] $\mathbb{R}^n$ is a real Hilbert space; the geometric notions of distance and angle defined above agree with ordinary [[Euclidean space|Euclidean geometry]] for this example.

\hypertarget{of_lebesgue_squareintegrable_functions_over_a_manifold}{}\subsubsection*{{Of Lebesgue square-integrable functions over a manifold}}\label{of_lebesgue_squareintegrable_functions_over_a_manifold}

The [[L-2-spaces|L-]] Hilbert spaces $L^2(\mathbb{R})$, $L^2([0,1])$, $L^2(\mathbb{R}^3)$, etc (real or complex) are very well known. In general, $L^2(X)$ for $X$ a [[measure space]] consists of the almost-everywhere defined functions $f$ from $X$ to the scalar field ($\mathbb{R}$ or $\mathbb{C}$) such that $\int |f|^2$ converges to a finite number, with functions identified if they are equal almost everywhere; we have $\langle f, g\rangle = \int \bar{f} g$, which converges by the Cauchy--Schwarz inequality. In the specific cases listed (and in general, when $X$ is a [[locally compact space|locally compact]] [[Hausdorff space]]), we can also get this space by completing the positive definite inner product space of compactly supported continuous functions.

\hypertarget{of_squareintegrable_halfdensities}{}\subsubsection*{{Of square-integrable half-densities}}\label{of_squareintegrable_halfdensities}

\begin{itemize}%
\item [[canonical Hilbert space of half-densities]]

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{bases}{}\subsubsection*{{Bases}}\label{bases}

A basic result is that abstractly, Hilbert spaces are all of the same type: every Hilbert space $H$ admits an orthonormal basis, meaning a [[subset]] $S \subseteq H$ whose inclusion map extends (necessarily uniquely) to an isomorphism

\begin{displaymath}
l^2(S) \to H
\end{displaymath}
of Hilbert spaces. Here $l^2(S)$ is the vector space consisting of those [[function]]s $x$ from $S$ to the scalar field such that

\begin{displaymath}
\|x\|^2 = \sum_{u: S} |x_u|^2
\end{displaymath}
converges to a finite number; this may also be obtained by completing the vector space of formal linear combinations of elements of $S$ with an inner product uniquely determined by the rule

\begin{displaymath}
\langle u, v \rangle = \delta_{u v} \qquad u, v \in S
\end{displaymath}
in which $\delta_{u v}$ denotes [[Kronecker delta]]. We thus have, in $l^2(S)$,

\begin{displaymath}
\langle x, y \rangle = \sum_{u: S} \bar{x}_u y_u .
\end{displaymath}
(This sum converges by the Cauchy--Schwarz inequality.)

In general, this result uses the [[axiom of choice]] (usually in the form of [[Zorn's lemma]] and [[excluded middle]]) in its proof, and is equivalent to it. However, the result for [[separable space|separable]] Hilbert spaces needs only [[dependent choice]] and so is [[constructive mathematics|constructive]] by most schools' standards. Even without dependent choice, explicit orthornormal bases for particular $L^2(X)$ can often be produced using [[approximation of the identity]] techniques, often in concert with a [[Gram-Schmidt process]].

In particular, all infinite-dimensional separable Hilbert spaces are abstractly isomorphic to $l^2(\mathbb{N})$.

\hypertarget{cauchyschwarz_inequality}{}\subsubsection*{{Cauchy--Schwarz inequality}}\label{cauchyschwarz_inequality}

The \emph{Schwarz inequality} (or \emph{Cauchy----Schwarz inequality}, etc) is very handy:

\begin{displaymath}
|\langle x, y \rangle| \leq \|x\| \|y\| .
\end{displaymath}
This is really two theorems (at least): an abstract theorem that the inequality holds in any Hilbert space, and concrete theorems that it holds when the inner product and norm are defined by the formulas used in the examples $L^2(X)$ and $l^2(S)$ above. The concrete theorems apply even to functions that don't belong to the Hilbert space and so prove that the inner product converges whenever the norms converge. (A somewhat stronger result is needed to conclude this convergence constructively; it may be found in Errett Bishop's book.)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[rigged Hilbert space]]


\item [[Hilbert C-star-module]], [[Hilbert bimodule]]


\item [[Kähler vector space]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Standard accounts of Hilbert spaces in [[quantum mechanics]] include

\begin{itemize}%
\item [[John von Neumann]], \emph{Mathematische Grundlagen der Quantenmechanik}. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.


\item [[George Mackey]], \emph{The Mathematical Foundations of Quamtum Mechanics} A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963


\item E. Prugoveki, \emph{Quantum mechanics in Hilbert Space}. Academic Press, 1971.



\end{itemize}
category: analysis

[[!redirects Hilbert spaces]]



\end{document}