Let 𝔽 be either or , and let p with p1. We define p to be the set of all sequences (ai)i0 in 𝔽 such that



We also define to be the set of all boundedPlanetmathPlanetmath (http://planetmath.org/BoundedInterval) sequences (ai)i0 with norm given by


By defining addition and scalar multiplication pointwise, p(𝔽) and (𝔽) have a natural vector spaceMathworldPlanetmath stucture. That the sum of two elements on p(𝔽) is again an element in p(𝔽) follows from Minkowski inequalityMathworldPlanetmath (see below). We can make p into a normed vector spacePlanetmathPlanetmath, by defining the norm as


The normed vector spaces and p for p1 are complete under these norms, making them into Banach spacesMathworldPlanetmath. Moreover, 2 is a Hilbert spaceMathworldPlanetmath under the inner productMathworldPlanetmath


where x¯ denotes the complex conjugateMathworldPlanetmath of x.

For p>1 the (continuousMathworldPlanetmath) dual spaceMathworldPlanetmath of p is q where 1p+1q=1, and the dual space of 1 is .


  1. 1.

    If a=(a0,a1,)p(𝔽) for 1p<, then limkak=0. (proof. (http://planetmath.org/ThenA_kto0IfSum_k1inftyA_kConverges))

  2. 2.

    For 1p<, p(𝔽) is separable, and (𝔽) is not separable.

  3. 3.

    Minkowski inequality. If a,bp(𝔽) where p1, then

Title ell^p
Canonical name Ellp
Date of creation 2013-03-22 12:19:03
Last modified on 2013-03-22 12:19:03
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 25
Author rspuzio (6075)
Entry type Definition
Classification msc 46B99
Classification msc 54E50
Related topic EllpXSpace
Defines 2