Lp space

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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 statistics, finance, and the engineering field of finite element analysis.

1 Motivation
2 spaces
3 Properties of spaces
4 Lp spaces
5 Special cases
6 Relation to spaces
7 Properties of Lp spaces
- 7.1 Dual spaces
- 7.2 Embeddings
8 Weighted Lp spaces
9 See also
10 Reference
11 External links

[edit] 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 $x=(x_1, x_2, \dots, x_n)$ is usually given by

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

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 $x=(x_1, x_2, \dots, x_n)$ . 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. For any p≥1, Rⁿ together with the p-norm just defined becomes a Banach space.

[edit] $\ell^p$ spaces

The above p-norm can be extended to vectors having an infinite number of components, yielding the space $\ell^p$ . 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, that being that the series on the right is not always convergent, so for example, the sequence made up of only ones, $(1, 1, 1, \dots),$ will have an infinite p-norm (length), no matter what p is. The space $\ell^p$ 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 $\ell^p$ spaces for 1≤p≤∞.

The p-norm thus defined on $\ell^p$ is indeed a norm, and $\ell^p$ together with this norm is a Banach space. The fully general L^p space, is obtained, as seen below, when one considers vectors not only with several components or with a countably infinite many components, but rather, vectors with arbitrarily many components, in other words, functions. Instead of using a sum to define the p-norm, one will use an integral.

[edit] Properties of $\ell^p$ spaces

The space $\ell^2$ is the only of the $\ell^p$ spaces 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 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.

[edit] L^p spaces

Let p be a positive real number and let (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

$\|f\|_p := \sqrt[p\!]{\int |f|^p\;\mathrm{d}\mu}<\infty.$

The vector sum of two functions is given by

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

and one checks that the sum of two p^th power integrable functions is again p^th power integrable. The scalar action on a function is given by

(λ f)(x) = λ f (x).

This vector space together with the function $\|\cdot\|_p$ is a seminormed complete vector space denoted by $\mathcal{L}^p(S, \mu)$ . To make it into a Banach space one considers the Kolmogorov quotient of this space, a standard procedure for spaces which are not T₀; one divides out the kernel of the norm. Thus we define $L^p(S, \mu) := \mathcal{L}^p(S, \mu) / \mathrm{ker}(\|\cdot\|_p)$ . 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, up to a set of measure zero, bounded. By identifying two such functions if they are equal almost everywhere, we get the set L^∞(S). For f in L^∞(S), we set

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

As before, we have

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

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

This definition is generalized in the context of Bochner spaces.

[edit] Special cases

The most important case is when p = 2; like the $\ell^2$ space, the space L² is the only Hilbert space of this class, 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.

[edit] Relation to $\ell^p$ spaces

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 a countable set S with the counting measure, the obtained 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.

[edit] Properties of L^p spaces

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.)

[edit] 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.

[edit] Embeddings

Suppose that the domain S has finite measure, and that 1 ≤ p ≤ q ≤ ∞. Then the bound

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

follows from Hölder's inequality. Hence, the space L^p is embedded in L^q.

It follows that, without any restriction on S, the embedding

$L^1_{\mathrm{loc}}(S) \subset L^p(S)$

holds, where the space on the left consists of the locally integrable functions on S.

[edit] Weighted L^p spaces

As before, consider a measure space $(S, \mathcal{F}, \mu)$ . Let $w : S \to [0, + \infty)$ be an absolutely integrable function, i.e. $w \in L^{1} (S, \mu)$ . 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