Lp space

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In mathematics, the L^p and ℓ^p spaces are spaces of p-power integrable functions, and corresponding sequence spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to Bourbaki (1987) they were first introduced by Riesz (1910). They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.

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

${\displaystyle \ (x_{1},x_{2},\dots ,x_{n})+(y_{1},y_{2},\dots ,y_{n})=(x_{1}+y_{1},x_{2}+y_{2},\dots ,x_{n}+y_{n}),}$

and the scalar action is given by

${\displaystyle \ \lambda (x_{1},x_{2},\dots ,x_{n})=(\lambda x_{1},\lambda x_{2},\dots ,\lambda x_{n}).}$

The length of a vector ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$ is usually given by

${\displaystyle \ \|x\|_{2}=\left(x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}\right)^{1/2}}$

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

${\displaystyle \ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p}}$

for any vector ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$. 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 (triangle inequality). For any p ≥ 1, Rⁿ together with the p-norm just defined becomes a Banach space.

The above p-norm can be extended to vectors having an infinite number of components, yielding the space ℓ^p. For ${\displaystyle \ x=(x_{1},x_{2},\dots ,x_{n},x_{n+1},\dots )}$ an infinite sequence of real (or complex) numbers, define the vector sum to be

${\displaystyle \ (x_{1},x_{2},\dots ,x_{n},x_{n+1},\dots )+(y_{1},y_{2},\dots ,y_{n},y_{n+1},\dots )=(x_{1}+y_{1},x_{2}+y_{2},\dots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\dots ),}$

while the scalar action is given by

${\displaystyle \ \lambda (x_{1},x_{2},\dots ,x_{n},x_{n+1},\dots )=(\lambda x_{1},\lambda x_{2},\dots ,\lambda x_{n},\lambda x_{n+1},\dots ).}$

Define the p-norm

${\displaystyle \ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}+|x_{n+1}|^{p}+\dots \right)^{1/p}.}$

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, ${\displaystyle (1,1,1,\dots ),}$ 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 ℓ^p grows larger. For example, the sequence

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

is not in ℓ¹, but it is in ℓ^p for p > 1, as the series

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

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

One also defines the ∞-norm as

${\displaystyle \ \|x\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dots ,|x_{n}|,|x_{n+1}|,\dots )}$

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

${\displaystyle \ \|x\|_{\infty }=\lim _{p\to \infty }\|x\|_{p}}$

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.

The space ℓ² is the only ℓ^p space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram identity ${\displaystyle \|x+y\|_{p}^{2}+\|x-y\|_{p}^{2}=2\|x\|_{p}^{2}+2\|y\|_{p}^{2}}$. Substituting two distinct unit vectors in for x and y directly shows that the identity is not true unless p = 2.

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

The space c₀ is defined as the space of all sequences converging to zero, with norm identical to ||x||_∞. The dual of c₀ is ℓ¹; the dual of ℓ¹ is ℓ^∞. For the case of natural numbers index set, the ℓ^p and c₀ are separable, with the sole exception of ℓ^∞. The dual of ℓ^∞ is the ba space.

The ℓ^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 ℓ^p is that it is not polynomially reflexive.

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

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

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

${\displaystyle (f+g)(x)=f(x)+g(x)\,}$

and, for a scalar λ,

${\displaystyle (\lambda f)(x)=\lambda f(x).\,}$

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 ${\displaystyle \|\cdot \|_{p}.}$

Thus the set of p^th power integrable functions, together with the function ${\displaystyle \|\cdot \|_{p}}$, is a seminormed vector space, which we denote by ${\displaystyle {\mathcal {L}}^{p}(S,\mu ).}$

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,

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

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:

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

As before, we have

${\displaystyle \|f\|_{\infty }=\lim _{p\to \infty }\|f\|_{p}}$

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

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

When the underlying measure space S is understood, L^p(S,μ) is often abbreviated L^p(μ), or just L^p. The above definitions generalize to Bochner spaces.

When p = 2; like the ℓ² 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 any L^p space by multiplication.

The ℓ^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 ℓ^p(S). For example, the space ℓ^p(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 ℓ^p(n), where n is the set with n elements, is Rⁿ with its p-norm as defined above.

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

${\displaystyle \kappa _{g}(f)=\int fg\;{\mbox{d}}\mu .}$

The mapping ${\displaystyle \kappa :g\mapsto \kappa _{g}}$ is a linear mapping from ${\displaystyle L^{q}(\mu )}$ into ${\displaystyle L^{p}(\mu )^{*}}$, which is an isometry onto its image by Hölder's inequality. It is also possible to show that any G ${\displaystyle \in L^{p}(\mu )^{*}}$ can be expressed this way: i.e., that κ is a continuous linear bijection of Banach spaces. By the open mapping theorem, it follows that κ is an isomorphism of Banach spaces.

Since the relationship 1/p + 1/q = 1 is symmetric, ${\displaystyle L^{p}(\mu )}$ is reflexive for these values of p: the natural monomorphism from ${\displaystyle L^{p}(\mu )}$ to ${\displaystyle L^{p}(\mu )^{**}}$ obtained by composing κ with the adjoint of its inverse

${\displaystyle L^{p}(\mu ){\overset {\kappa }{\to }}L^{q}(\mu )^{*}{\overset {\,\,(\kappa ^{-1})^{*}}{\longrightarrow }}L^{p}(\mu )^{**}}$

is onto, that is, it is an isomorphism of Banach spaces.

If the measure μ on S is sigma-finite, then the dual of L¹(μ) is isomorphic to L^∞(μ). 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.

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 the following:

1. Let 1 ≤ p < q ≤ ∞. L^p(S) is contained in L^q(S) iff S does not contain sets of arbitrarily large measure, and
2. Let 1 ≤ p < q ≤ ∞. L^q(S) is contained in L^p(S) iff S does not contain sets of arbitrarily small measure.

In particular, if the domain S has finite measure, the bound (a consequence of Hölder's inequality)

${\displaystyle \ \|f\|_{p}\leq \mu (S)^{(1/p)-(1/q)}\|f\|_{q}}$

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

${\displaystyle L^{p}}$ spaces are widely used in mathematics and applications.

The Fourier transform for the real line (resp. for periodic functions, cf. Fourier series) maps ${\displaystyle L^{p}(\mathbb {R} )}$ to ${\displaystyle L^{q}(\mathbb {R} )}$ (resp. ${\displaystyle L^{p}(\mathbb {T} )}$ to ${\displaystyle \ell ^{q}}$), where ${\displaystyle 1\leq p\leq 2}$ and 1/p+1/q=1. This is a consequence of the Riesz-Thorin theorem, and is made precise with the Hausdorff-Young inequality.

By contrast, if p>2, the Fourier transform does not map into ${\displaystyle L^{q}}$.

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces ${\displaystyle L^{2}}$ and ${\displaystyle \ell ^{2}}$ are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ${\displaystyle \ell ^{2}(E)}$, where ${\displaystyle E}$ is a set with an appropriate cardinality.

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of ${\displaystyle L^{p}}$ metrics, and measures of central tendency can be characterized as solutions to variational problems.

Let (S, μ) be a measure space, and f a measurable function with real or complex values on S. The distribution function of f is defined for t > 0 by

${\displaystyle \lambda _{f}(t)=\mu \left\{x\in S\mid |f(x)|>t\right\}.}$

If f is in L^p(S) for some p with 1 ≤ p <∞, then by Markov's inequality,

${\displaystyle \lambda _{f}(t)\leq {\frac {\|f\|_{p}^{p}}{t^{p}}}.}$

A function f is said to be in the space weak L^p(S), or L^p,w(S), if there is a constant C > 0 such that, for all t > 0,

${\displaystyle \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}.}$

The best constant C for this inequality is the L^p,w-norm of f, and is denoted by

${\displaystyle \|f\|_{p,w}=\inf \left\{C\mid \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}\quad \forall t>0\right\}.}$

The L^p,w-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for f in L^p(S),

${\displaystyle \|f\|_{p,w}\leq \|f\|_{p},}$

and in particular L^p(S) ⊂ L^p,w(S).

A major result on L^p,w-spaces is Marcinkiewicz interpolation, which has broad applications to harmonic analysis and the study of singular integrals.

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

${\displaystyle \ \nu (A):=\int _{A}w(x)\,\mathrm {d} \mu (x),}$

or, in terms of the Radon-Nikodym derivative,

${\displaystyle \ w={\frac {\mathrm {d} \nu }{\mathrm {d} \mu }}.}$

The norm for ${\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}$ is explicitly

${\displaystyle \ \|u\|_{L^{p}(S,w\,\mathrm {d} \mu )}:=\left(\int _{S}w(x)|u(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}.}$

• Hardy space
• Hölder mean
• Hölder space
• Root mean square
• Locally integrable function (${\displaystyle L_{\text{loc}}^{1}}$)

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• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3540136279.
• DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN 3-7643-4231-5.
• Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience.
• Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag.
• Riesz, F (1910), "Untersuchungen über Systeme integrierbarer Funktionen", Mathematische Annalen, 69: 449–497.

• "Proof that L^p spaces are complete". PlanetMath.