Lp space

In mathematics, the L^p and l^p 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.

L^p spaces

Given a positive real number p and a measure space S and 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. Identifying two such functions if they are equal almost everywhere, we obtain the set L^p(S). For f in L^p(S), we define

$\|f\|_{p}=\left(\int |f(x)|^{p}\;dx\right)^{1/p}.$

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

$\|f\|_{\infty }=\inf\{C\geq 0:|f(x)|\leq C{\mbox{ almost everywhere}}\}.$

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 one chooses S to be the unit interval [0,1] with the Lebesgue measure, then the corresponding L^p space is denoted by L^p([0,1]). For p < ∞ it consists of all functions f : [0,1] → C (or R) so that |f|^p has a finite integral, again with functions that are equal almost everywhere being identified. The space L^∞([0,1]) consists of all measurable functions f : [0,1] → C (or R) such that |f| is bounded almost everywhere, with functions that are equal almost everywhere being identified. The spaces L^p(R) are defined similarly.

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 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

$G(f)=\int f(x)^{*}\,g(x)\,{\mbox{d}}x,\qquad {\mbox{for all }}f\in L^{p}.$

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, even for sigma-finite measure spaces, the dual of L^∞ is usually much bigger than L¹ and is isometric to the $ba(\Sigma )$ space.

If 0 < p < 1, then L^p can be defined as above, but it won't be a Banach space as 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.

l^p spaces

The $\ell ^{p}$ spaces are of a similiar concept. They can be treated as a special case of the $L^{p}$, when the measure used in the integration in the definiton is the counting measure instead of the aboveused Lebesgue measure and the measure space is discrete. Thus,

$\ell ^{p}(S)$

(1 ≤ p ≤ ∞) is defined as set of sequences

$x=\{x_{i}\}$, $i\in S$

for which

$\|x\|_{p}<\infty ,$

where

$\|x\|_{p}=\left(\sum _{i\in S}|x_{i}|^{p}\right)^{1/p}.$

As with $L^{p}$ spaces, the $\|x\|_{\infty }$ is defined as

$\sup _{i\in S}|x_{i}|.$

If S is the set of natural numbers, the space is usually denoted as $\ell ^{p}$ (ie., without the space indication).

Closely connected to $\ell ^{p}$ is the c₀, which is defined as space of all sequences declining to zero, with norm identical to $\|x\|_{\infty }$.

Properties

The space $\ell ^{2}$ is a Hilbert space (and no other $\ell ^{p}$ is).

The $\ell ^{p}$, 1 < p < ∞ spaces are reflexive: $(\ell ^{p})^{*}=\ell ^{q}$, where (1/p) + (1/q) = 1. If the index set S is infinite, then so are $\ell ^{1}$, $\ell ^{\infty }$, and c₀.

The dual of c₀ is $\ell ^{1}$; the dual of $\ell ^{1}$ is $\ell ^{\infty }$. For the case of natural numbers index set, the $l^{p}$ and c₀ are separable, with the sole exception of $\ell ^{p}$.

The $\ell ^{p}$ spaces can be found embeded into most of Banach spaces. The question whether all Banach spaces has such embeding was answered (negativelly) by B. S. Tsirelson's construction of Tsirelson space in 1974.

Except for the trivial finite case, none of the $l^{p}$ space is polynomially reflexive. This makes such spaces very rare.