Lp space

The correct title of this article is L^p space. It features superscript or subscript characters that are substituted or omitted because of technical limitations.

In mathematics, the L^p and $\ell^p$ 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.

L^p spaces have applications in physics, statistics, finance, engineering, etc.

Motivation

Consider the real vector space Rⁿ. The sum of vectors in Rⁿ is given by

$\ (x_1, x_2, \dots, x_n) + (y_1, y_2, \dots, y_n) = (x_1+y_1, x_2+y_2, \dots, x_n+y_n),$

and the scalar action is given by

$\ \lambda(x_1, x_2, \dots, x_n)=(\lambda x_1, \lambda x_2, \dots, \lambda x_n).$

The length of a vector $x = (x 1, x 2,..., x n)$ is usually given by

$\ \|x\|_2=\left(x_1^2+x_2^2+\dots+x_n^2\right)^{1/2}$

but this is by no means the only way of defining length. If p is a real number, p ≥ 1, define

$\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dots+|x_n|^p\right)^{1/p}$

for any vector $x = (x 1, x 2,..., x n)$ . It turns out that this definition indeed satisfies the properties of a "length function" (or norm), which are that only the zero vector has zero length, the length of the vector changes (modulus-)linearly when we multiply it by a scalar, and the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality). For any p ≥ 1, Rⁿ together with the p-norm just defined becomes a Banach space.

ℓ^p spaces

The above p-norm can be extended to vectors having an infinite number of components, yielding the space ℓ^p. For $\ x=(x_1, x_2, \dots, x_n, x_{n+1},\dots)$ an infinite sequence of real (or complex) numbers, define the vector sum to be

$\ (x_1, x_2, \dots, x_n, x_{n+1},\dots)+(y_1, y_2, \dots, y_n, y_{n+1},\dots)=(x_1+y_1, x_2+y_2, \dots, x_n+y_n, x_{n+1}+y_{n+1},\dots),$

while the scalar action is given by

$\ \lambda(x_1, x_2, \dots, x_n, x_{n+1},\dots) = (\lambda x_1, \lambda x_2, \dots, \lambda x_n, \lambda x_{n+1},\dots).$

Define the p-norm

$\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dots+|x_n|^p+|x_{n+1}|^p+\dots\right)^{1/p}.$

Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, $(1,1,1,...),$ will have an infinite p-norm (length) for every finite p ≥ 1. The space ℓ^p is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite.

One can check that as p increases, the set $\ell^p$ grows larger. For example, the sequence

$\ \left(1, \frac{1}{2}, \dots, \frac{1}{n}, \frac{1}{n+1},\dots\right)$

is not in $\ell^1$ , but it is in $\ell^p$ for p>1, as the series

$\ 1^p+\frac{1}{2^p} + \dots + \frac{1}{n^p} + \frac{1}{(n+1)^p}\dots$

diverges for p=1 (the harmonic series), but is convergent for p>1.

One also defines the ∞-norm as

$\ \|x\|_\infty=\sup(|x_1|, |x_2|, \dots, |x_n|,|x_{n+1}|, \dots)$

and the corresponding space $\ell^\infty$ of all bounded sequences. It turns out that

$\ \|x\|_\infty=\lim_{p\to\infty}\|x\|_p$

if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider ℓ^p spaces for 1≤p≤∞.

The p-norm thus defined on ℓ^p is indeed a norm, and ℓ^p together with this norm is a Banach space. The fully general L^p space is obtained — as seen below — by considering vectors, not only with finitely or countably-infinitely many components, but with arbitrarily many components; in other words, functions. An integral instead of a sum is used to define the p-norm.

Properties of ℓ^p spaces

The space $\ell^2$ is the only $\ell^p$ space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram identity $\|x+y\|_p^2 + \|x-y\|_p^2= 2\|x\|_p^2 + 2\|y\|_p^2$ . Direct substitution with unit vectors results in a counter example.

The $\ell^p$ , 1 < p < ∞ spaces are reflexive: $(\ell^p)^*=\ell^q$ , where (1/p) + (1/q) = 1.

The dual of c₀ is $\ell^1$ ; the dual of $\ell^1$ is $\ell^\infty$ . For the case of natural numbers index set, the $\ell^p$ and c₀ are separable, with the sole exception of $\ell^{\,\infty}$ . Here, c₀ is defined as the space of all sequences converging to zero, with norm identical to ||x||_∞.

The $\ell^p$ spaces can be 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 $\ell^p$ is that it is not polynomially reflexive.

L^p spaces

Let 1 ≤ p < ∞ and (S, μ) be a measure space. Consider the set of all measurable functions from S to C (or R) whose absolute value raised to the p-th power has a finite Lebesgue integral, or equivalently, that

$\|f\|_p := \left({\int |f|^p\;\mathrm{d}\mu}\right)^{1/p}<\infty.$

The set of such functions form a vector space, with the following natural operations:

$(f+g)(x)=f(x)+g(x) \,$

and, for a scalar λ,

$(\lambda f)(x) = \lambda f(x). \,$

That the sum of two p^th power integrable functions is again p^th power integrable follows from the inequality |f + g|^p ≤ 2^p (|f|^p + |g|^p). In fact, more is true. Minkowski's inequality says the triangle inequality holds for

$\| \cdot \|_p.$

Thus the set of p^th power integrable functions, together with the function ||·||_p, a seminormed vector space, which we denote by

$\mathcal{L}^p(S, \mu).$

This can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the kernel of ||·||_p. Since ||f||_p = 0 if and only if f = 0 almost everywhere, in the quotient space two functions f and g are identified if f = g almost everywhere. The resulting normed vector space is, by definition,

$L^p(S, \mu) := \mathcal{L}^p(S, \mu) / \mathrm{ker}(\|\cdot\|_p) .$

For p = ∞, the space L^∞(S, μ) is defined as follows. We start with the set of all measurable functions from S to C (or R) which are essentially bounded, i.e. bounded up to a set of measure zero. Again two such functions are identified if they are equal almost everywhere. Denote this set by L^∞(S, μ). For f in L^∞(S, μ), its essential supremum serves as an appropriate norm:

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

As before, we have

$\|f\|_\infty=\lim_{p\to\infty}\|f\|_p$

if f ∈ L^∞(S) ∩ L^q(S) for some q < ∞.

For 1 ≤ p ≤ ∞, L^∞(S, μ) is a Banach space. Completeness can be checked using the convergence theorems for Lebesgue integrals.

The above definitions generalize to Bochner spaces.

Special cases

When p = 2; like the $\ell^2$ space, the space L² is the only Hilbert space of this class. The additional inner product structure allows for a richer theory, with applications to, for instance Fourier series and quantum mechanics.

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. An element of L^∞ defines a bounded operator on the Hilbert space L² by multiplication.

The $\ell^p$ spaces (1≤p≤∞) are a special case of L^p spaces, when the set S is the positive integers, and the measure used in the integration in the definition is a counting measure. More generally, if one considers any set S with the counting measure, the resulting L^p space is denoted $\ell^p(S)$ . For example, the space $\ell^p(\mathbb Z)$ is the space of all sequences indexed by the integers, and when defining the p-norm on such a space, one sums over all the integers. The space $\ell^p(n)$ , where n is the set with n elements, is Rⁿ with its p-norm as defined above.

Properties of L^p spaces

Dual spaces

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

$G(f) = \int \bar{f} g \;\mbox{d}\mu$

(where $\bar{f}$ means the complex conjugate). It is possible to show that any G $\in (L^p)^*$ can be expressed this way. 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 || · ||_p does not satisfy the triangle inequality in this case, and hence it defines only a quasi-norm. 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.

Embeddings

Colloquially, if 1 ≤ p ≤ q ≤ ∞, L^p(S) contains functions that are more locally singular while elements of L^q(S) can be more spread out. Consider the Lebesgue measure on the half line (0, ∞). A continuous function in L¹ might blow up near 0 but must decay sufficiently fast toward infinity. On the other hand, continuous functions in L^∞ need not decay at all but no blow-up is allowed. The precise technical result is following:

L^p(S) is not contained in L^q(S) iff S contains sets of arbitrarily small measure, and

L^q(S) is not contained in L^p(S) iff S contains sets of arbitrarily large measure. In particular, if the domain S has finite measure, the bound (a consequence of Hölder's inequality)

$\ \|f\|_p \le \mu(S)^{(1/p)-(1/q)} \|f\|_q$

means the space L^q is continuously embedded in L^p.

Weighted L^p spaces

As before, consider a measure space $(S, \mathcal{F}, \mu)$ . Let $w : S \to[0, + \infty)$ be a measurable function. The $w$ -weighted L^p space is defined as $L^{p} (S, w \, \mathrm{d} \mu)$ , where $w \, \mathrm{d} \mu$ means the measure $ν$ defined by