Lp space

The title of this article is incorrect because of technical limitations. The correct title is L^p space..

In mathematics, the L^p and <math>\ell^p<math> spaces are spaces of p-power integrable functions, and corresponding sequence spaces. They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. See also root mean square, Hardy space.

Contents

1 Lp spaces 2 Special cases 3 Further properties 4 l p spaces 5 Properties

L^p spaces

Let p be a positive real number and let (S, μ) be a measure space. We define

<math>\|\cdot\|_p : \mathcal{B}(S, \mathbf{C}) \to \mathbf{R} : f \mapsto \|f\|_p := \sqrt[p\!]{\int |f|^p\;\mathrm{d}\mu}.<math>

Consider the set of all measurable functions from S to C (or R) whose absolute value to the p-th power has a finite Lebesgue integral. It is a seminormed complete vector space and we denote it by <math>\mathcal{L}^p(S, \mu)<math>. To make it into a Banach space we need to divide out the kernel of the norm. Thus we define <math>L^p(S, \mu) := \mathcal{L}^p(S, \mu) / \mathrm{ker}(\|\cdot\|_p)<math>. This means we are identifying two functions if they are equal almost everywhere. The space L^∞(S), while related, is defined differently. We start with the set of all measurable functions from S to C (or R) which are bounded almost everywhere. By identifying two such functions if they are equal almost everywhere, we get the set L^∞(S). For f in L^∞(S), we set

<math>\|f\|_\infty := \inf \{ C\ge 0 : |f(x)| \le C \mbox{ for almost every } x\}.<math>

Special cases

The most important case is when p = 2; the space L² is a Hilbert space, having major applications to Fourier series and quantum mechanics, as well as other fields.

If we use complex-valued functions, the space L^∞ is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra, since any element of L^∞ defines an operator on the Hilbert space L² by pointwise multiplication.

Further properties

If 1 ≤ p ≤ ∞, then the Minkowski inequality, proved using Hölder's inequality, establishes the triangle inequality in L ^p(S). Using the convergence theorems for the Lebesgue integral, one can then show that L ^p(S) is complete and hence is a Banach space. (Here it is crucial that the Lebesgue integral is employed, and not the Riemann integral.)

The dual space (the space of all continuous linear functionals) of L ^p for 1 < p < ∞ has a natural isomorphism with L ^q where q is such that 1/p + 1/q = 1, which associates g ∈ L ^q with the functional G defined by

<math> G(f) = \int f^*g \; \mbox{d}\mu<math>

Since the relationship 1/p + 1/q = 1 is symmetric, L ^p is reflexive for these values of p: the natural monomorphism from L ^p to (L ^p)^** is onto, that is, it is an isomorphism of Banach spaces.

If the measure on S is sigma-finite, then the dual of L¹(S) is isomorphic to L^∞(S). However, except in rather trivial cases, the dual of L^∞ is much bigger than L¹. Elements of (L^∞)^* can be identified with bounded signed finitely additive measures on S in a construction similar to the ba space.

If 0 < p < 1, then L ^p can be defined as above, but it is not a Banach space since the triangle inequality does not hold in general. However, we can still define a metric by setting d(f, g) = (||f − g||_p)^p. The resulting metric space is complete, and L ^p for 0 < p < 1 is the prototypical example of an F-space that is not locally convex. The map sending f to ||f||_p is a quasi-norm, and L ^p is a quasi-Banach space, that is, a complete quasi-normed vector space.

l ^p spaces

The <math>\ell^p<math> spaces are a special case of L ^p spaces, in which the measure used in the integration in the definition is a counting measure and the measure space S is discrete. Thus, for 0 < p < ∞, <math>\ell^p<math>(S) is defined as the set of sequences x = {x _i}, i in S, for which the quantity

<math>\|x\|_p = \sqrt[{}^p]{\sum_{i\in S} |x_i|^p}<math>

is finite. As with L ^p spaces, the sup norm ||x||_∞ is defined as

<math>\|x\|_{\infty}=\sup_{i\in S} |x_i|.<math>

If S is the set of natural numbers, the space <math>\ell^p(S)<math> is usually denoted as <math>\ell^p<math> (i.e., the measure space is suppressed).

Closely connected to <math>\ell^p<math> is c₀, which is defined as the space of all sequences declining to zero, with norm identical to ||x||_∞.

Properties

The space <math>\ell^2<math> is a Hilbert space (and no other <math>\ell^p<math> is).

The <math>\ell^p<math>, 1 < p < ∞ spaces are reflexive: <math>(\ell^p)^*=\ell^q<math>, where (1/p) + (1/q) = 1. If the index set S is infinite, then <math>\ell^1<math>, <math>\ell^\infty<math>, and c₀ are not reflexive.

The dual of c₀ is <math>\ell^1<math>; the dual of <math>\ell^1<math> is <math>\ell^\infty<math>. For the case of natural numbers index set, the <math>\ell^p<math> and c₀ are separable, with the sole exception of <math>\ell^\infty<math>.

The <math>\ell^p<math> spaces can be found embedded into many Banach spaces. The question of whether all Banach spaces have such an embedding was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974.

Except for the trivial finite case, an unusual feature of <math>\ell^p<math> is that it is not polynomially reflexive.