p(X) space



0.0.1 Definition of p(X)

Let p be a real number such that 1p<.

Let X be a set and let μ be the counting measure on X, defined on the σ-algebra (http://planetmath.org/SigmaAlgebra) 𝔅 of all subsets of X. The p(X) space is a particular of a Lp-space (http://planetmath.org/LpSpace), defined as


Thus, the p(X) space consists of all functions f:X such that


Of course, for the above sum to be finite one must necessarily have f(x)0 only for a countableMathworldPlanetmath number of xX (see this entry (http://planetmath.org/SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable)).

0.0.2 Properties

  • By the corresponding property on Lp-spaces, the space p(X) is a Banach spaceMathworldPlanetmath and its norm amounts to

  • By the corresponding property on L2-spaces (http://planetmath.org/L2SpacesAreHilbertSpaces), the space 2(X) is a Hilbert spaceMathworldPlanetmath and its inner productMathworldPlanetmath amounts to


0.0.3 Nonseparability of p(X) for uncountable X

- The space p(X) is separable if and only if X is a countable set. Moreover, p(X) admits a Schauder basisMathworldPlanetmath if and only if X is countable.

A Schauder basis for p(X), when it exists, can be just the set of functions {δx0:x0X} defined by


0.0.4 Orthonormal basis of 2(X)

The set of functions {δx0:x0X} is an orthonormal basisMathworldPlanetmath of 2(X). Hence, the dimensionPlanetmathPlanetmath (http://planetmath.org/OrthonormalBasis) of 2(X) is given by the cardinality of X (as all orthonormal bases have the same cardinality).

It can be shown that all Hilbert spaces are isometrically isomorphic (hence, preserving the inner product) to a 2(X) space, for a suitable set X.

Title p(X) space
Canonical name ellpXSpace
Date of creation 2013-03-22 17:55:59
Last modified on 2013-03-22 17:55:59
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Definition
Classification msc 46E30
Classification msc 46B26
Classification msc 28B15
Synonym p(X)
Synonym p(X)-space
Related topic Lp
Related topic ClassificationOfHilbertSpaces
Related topic RieszFischerTheorem
Defines 2(X)
Defines 2(X) space
Defines p(X) is nonseparable iff X is uncountable
Defines orthonormal basis of 2(X) have the cardinality of X