$L^2$-spaces are Hilbert spaces

Let $(X,𝔅,\mu )$ 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 $f:X⟶ℂ$ 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 $⟨\cdot ,\cdot ⟩$ defined by

$$⟨f,g⟩={\int }_{X}f\overline{g}𝑑\mu$$

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 ${\int }_X{|f|}^2𝑑\mu =0$, 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

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

• 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	2013-03-22 17:32:25
Last modified on	2013-03-22 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