L2-spaces are Hilbert spaces

Let (X,𝔅,μ) be a measure spaceMathworldPlanetmath. Let L2(X) denote the L2-space (http://planetmath.org/LpSpace) associated with this measure space, i.e. L2(X) consists of measurable functionsMathworldPlanetmath f:X such that


identified up to equivalence almost everywhere.

It is known that all Lp-spaces (http://planetmath.org/LpSpace), with 1p, are Banach spacesMathworldPlanetmath with respect to the Lp-norm (http://planetmath.org/LpSpace) p. For L2(X) we can say more:

Theorem - L2(X) is an Hilbert SpaceMathworldPlanetmath with respect to the inner productMathworldPlanetmath , defined by



Sesquilinearity follows from the linearity of the Lebesgue integralMathworldPlanetmath (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions) (that is, the inner product defined above is linear in the first argumentMathworldPlanetmath and conjugate linear in the second one). The conjugate symmetry is evident.

Positive definiteness holds by construction: If X|f|2𝑑μ=0, then |f|2 (and therefore f) is zero almost everywhere, thus the equivalence classMathworldPlanetmathPlanetmath of f is the equivalence class of the zero function (which is the additivePlanetmathPlanetmath neutral element of the space).

Completeness is proved for the general case of Lp-spaces in this article (http://planetmath.org/ProofThatLpSpacesAreComplete).

0.0.1 Remarks

  • The spaces n or n with the usual inner product are particular examples of L2(X), choosing X={1,,n} with the counting measure.

  • Choosing appropriate spaces X it can be shown that all Hilbert spaces are isometrically isomorphic to a L2-space.

Title L2-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