${L}^{2}$spaces are Hilbert spaces
Let $(X,\U0001d505,\mu )$ be a measure space^{}. Let ${L}^{2}(X)$ denote the ${L}^{2}$space (http://planetmath.org/LpSpace) associated with this measure space, i.e. ${L}^{2}(X)$ consists of measurable functions^{} $f:X\u27f6\u2102$ such that
$$ 
identified up to equivalence almost everywhere.
It is known that all ${L}^{p}$spaces (http://planetmath.org/LpSpace), with $1\le p\le \mathrm{\infty}$, are Banach spaces^{} with respect to the ${L}^{p}$norm (http://planetmath.org/LpSpace) $\parallel \cdot {\parallel}_{p}$. For ${L}^{2}(X)$ we can say more:
Theorem  ${L}^{2}(X)$ is an Hilbert Space^{} with respect to the inner product^{} $\u27e8\cdot ,\cdot \u27e9$ defined by
$$\u27e8f,g\u27e9={\int}_{X}f\overline{g}\mathit{d}\mu $$ 
Proof:
Sesquilinearity follows from the linearity of the Lebesgue integral^{} (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions) (that is, the inner product defined above is linear in the first argument^{} and conjugate linear in the second one). The conjugate symmetry is evident.
Positive definiteness holds by construction: If ${\int}_{X}{f}^{2}\mathit{d}\mu =0$, then ${f}^{2}$ (and therefore $f$) is zero almost everywhere, thus the equivalence class^{} of $f$ is the equivalence class of the zero function (which is the additive^{} neutral element of the space).
Completeness is proved for the general case of ${L}^{p}$spaces in this article (http://planetmath.org/ProofThatLpSpacesAreComplete).$\mathrm{\square}$
0.0.1 Remarks

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The spaces ${\u2102}^{n}$ or ${\mathbb{R}}^{n}$ with the usual inner product are particular examples of ${L}^{2}(X)$, choosing $X=\{1,\mathrm{\dots},n\}$ with the counting measure.

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Choosing appropriate spaces $X$ it can be shown that all Hilbert spaces are isometrically isomorphic to a ${L}^{2}$space.
Title  ${L}^{2}$spaces are Hilbert spaces 

Canonical name  L2spacesAreHilbertSpaces 
Date of creation  20130322 17:32:25 
Last modified on  20130322 17:32:25 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  23 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 46C05 
Synonym  square integrable functions form an Hilbert space 
Related topic  LpSpace 
Related topic  HilbertSpace 
Related topic  MeasureSpace 
Related topic  BanachSpace 
Related topic  RieszFischerTheorem 
Defines  linear space of square integrable functions 
Defines  sequilinearity 