# $L^{2}$-spaces are Hilbert spaces

Let $(X,\mathfrak{B},\mu)$ 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 $f:X\longrightarrow\mathbb{C}$ such that

 $\|f\|_{2}:=\left(\int_{X}|f|^{2}d\mu\right)^{\frac{1}{2}}<\infty$

identified up to equivalence almost everywhere.

It is known that all $L^{p}$-spaces (http://planetmath.org/LpSpace), with $1\leq p\leq\infty$, are Banach spaces with respect to the $L^{p}$-norm (http://planetmath.org/LpSpace) $\;\|\cdot\|_{p}$. For $L^{2}(X)$ we can say more:

$L^{2}(X)$ is an Hilbert Space with respect to the inner product $\langle\cdot,\cdot\rangle$ defined by

 $\langle f,g\rangle=\int_{X}f\overline{g}\;d\mu$

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 $\int_{X}|f|^{2}d\mu=0$, 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).$\square$

## 0.0.1 Remarks

• The spaces $\mathbb{C}^{n}$ or $\mathbb{R}^{n}$ with the usual inner product are particular examples of $L^{2}(X)$, choosing $X=\{1,\dots,n\}$ 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 L2spacesAreHilbertSpaces 2013-03-22 17:32:25 2013-03-22 17:32:25 asteroid (17536) asteroid (17536) 23 asteroid (17536) Theorem msc 46C05 square integrable functions form an Hilbert space LpSpace HilbertSpace MeasureSpace BanachSpace RieszFischerTheorem linear space of square integrable functions sequilinearity