-spaces are Hilbert spaces
identified up to equivalence almost everywhere.
It is known that all -spaces (http://planetmath.org/LpSpace), with , are Banach spaces with respect to the -norm (http://planetmath.org/LpSpace) . For we can say more:
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 , then (and therefore ) is zero almost everywhere, thus the equivalence class of 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 -spaces in this article (http://planetmath.org/ProofThatLpSpacesAreComplete).
The spaces or with the usual inner product are particular examples of , choosing with the counting measure.
Choosing appropriate spaces it can be shown that all Hilbert spaces are isometrically isomorphic to a -space.
|Title||-spaces are Hilbert spaces|
|Date of creation||2013-03-22 17:32:25|
|Last modified on||2013-03-22 17:32:25|
|Last modified by||asteroid (17536)|
|Synonym||square integrable functions form an Hilbert space|
|Defines||linear space of square integrable functions|