Lp space

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.

Motivation

Illustrations of unit circles in different p-norms.

Unit circle (superellipse) in p=3/2 norm.

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 = (x₁, x₂, ..., x_n) is usually given by the Euclidean norm

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

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

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

(so the L² norm is the familiar Euclidean norm, while the distance in the L¹ norm is known as the Manhattan distance).

One also extends this to p = ∞ via

$\ \|x\|_\infty=\max \left\{|x_1|, |x_2|, \ldots, |x_n|\right\}$

which is in fact the limit of the p norms for finite p. The L^∞ norm is also known as the uniform norm.

It turns out that for all p ≥ 1 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 is positive homogeneous with respect to multiplication 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 L^p norm (or simply p-norm) just defined becomes a Banach space.

When 0 < p < 1

Astroid, unit circle in p=2/3 norm

In Rⁿ for n > 1, the formula

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

makes sense for 0 < p < 1, though the resulting function does not define a norm, because it violates the triangle inequality. However, the function

$d_p(x,y) = \sum_{i=1}^n |x_i-y_i|^p$

defines a metric. The metric space (Rⁿ, d_p) is denoted by ℓ_n^p.

Although the p-unit ball B_n^p around the origin in this metric is "concave", the topology defined on Rⁿ by the metric d_p is the usual vector space topology of Rⁿ, hence ℓ_n^p is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ℓ_n^p is to denote by C_p(n) the smallest constant C such that the multiple C B_n^p of the p-unit ball contains the convex hull of B_n^p, equal to B_n¹. The fact that C_p(n) = n^1/p – 1 tends to infinity with n (for fixed p < 1) announces the fact that the infinite-dimensional sequence space ℓ^p defined below, is no longer locally convex.

ℓ^p spaces

The above p-norm can be extended to vectors having an infinite number of components, yielding the space ℓ^p. This contains as special cases:

ℓ¹, the space of absolutely convergent series,
ℓ², the space of square-summable sequences, which is a Hilbert space, and
ℓ^∞, the space of bounded sequences.

The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. Explicitly, 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

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

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 ℓ^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 ℓ¹, but it is in ℓ^p for p > 1, as the series

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

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 ℓ^∞ 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 and the space c₀

See also: c space

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 $\|x+y\|_p^2 + \|x-y\|_p^2= 2\|x\|_p^2 + 2\|y\|_p^2$ . Substituting two distinct unit vectors in for x and y directly shows that the identity is not true unless p = 2.

If 1 < p < ∞, then the (continuous) dual space of ℓ^p is ℓ^q, where q is the Hölder conjugate of p: 1/p + 1/q = 1. Symbolically, (ℓ^p)^* = ℓ^q. As a consequence ℓ^p is reflexive, because (ℓ^p)^** = (ℓ^q)^* = ℓ^p (for a more detailed study of duality, see the section "Dual of L^p" below, that also applies here).

The space c₀ is defined as the space of all sequences converging to zero, with norm identical to ||x||_∞. It is a closed subspace of ℓ^∞, hence a Banach space. 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 spaces c₀ and ℓ^p (for 1 ≤ p < ∞) have a canonical Schauder basis {e_i | i = 1, 2,…}, where e_i is the sequence which is zero but for a 1 in the i^th entry.

The ℓ^p spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ^p or of c₀, was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space of ℓ¹, has an answer in the affirmative (easy and old).

Except for the trivial finite dimensional case, an unusual feature of ℓ^p is that it is not polynomially reflexive.

L^p spaces

Let 1 ≤ p < ∞ and (S, Σ, 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

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

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

$(f+g)(x)=f(x)+g(x), \ \ \ \text{and} \ \ \ (\lambda f)(x) = \lambda f(x) \,$

for every scalar λ.

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 || . ||_p. Thus the set of p^th power integrable functions, together with the function || . ||_p, is a seminormed vector space, which is denoted 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 for any measurable function f, we have that ||f||_p = 0 if and only if f = 0 almost everywhere, the kernel of || . ||_p does not depend upon p,

$N�:= \mathrm{ker}(\|\cdot\|_p) = \{f�: f = 0 \ \mu\text{-almost everywhere} \}.$

In the quotient space, two functions f and g are identified if f = g almost everywhere. The resulting normed vector space is, by definition,

$L^p(S, \mu)�:= \mathcal{L}^p(S, \mu) / N .$

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, μ) ∩ L^q(S, μ) for some q < ∞.

For 1 ≤ p ≤ ∞, L^p(S, μ) is a Banach space. The fact that L^p is complete is often referred to as Riesz-Fischer theorem. 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.

Special cases

When p = 2; like the ℓ² space, the space L² is the only Hilbert space of this class. In the complex case, the inner product on L² is defined by

$\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x).$

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in L² are sometimes called square-integrable or square-summable, but sometimes these terms are reserved for functions which are square-integrable in some other sense (such as in the sense of a Riemann integral).^[1]

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 S is the set N of positive integers, and the measure μ is the counting measure on N. 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. As any Hilbert space, every space L² is linearly isometric to a suitable ℓ²(I), where the cardinality of the set I is the cardinality of an arbitrary Hilbertian basis for this particular L².

Properties of L^p spaces

Dual spaces

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

$\kappa_p(g)�: f \in L^p(\mu) \mapsto \int f g \, \mathrm{d}\mu.$

The fact that κ_p(g) is well defined and continuous follows from Hölder's inequality. The mapping κ_p is a linear mapping from L^q(μ) into L^p(μ)^∗, which is an isometry by the extremal case of Hölder's inequality. It is also possible to show (for example with the Radon-Nikodym theorem) that any G ∈ L^p(μ)^∗ can be expressed this way: i.e., that κ_p is onto. Since κ_p is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say shortly that L^q "is" the dual of L^p.

When 1 < p < ∞, the space L^p(μ) is reflexive. Let κ_p be the above map and let κ_q be the corresponding linear isometry from L^p(μ) onto L^q(μ)^∗. The map

$j_p�: L^p(\mu) \overset{\kappa_q}{\to} L^q(\mu)^* \overset{\,\,(\kappa_p^{-1})^*}{\longrightarrow} L^p(\mu)^{**}$

from L^p(μ) to L^p(μ)^∗∗, obtained by composing κ_q with the transpose (or adjoint) of the inverse of κ_p, coincides with the canonical embedding J of L^p(μ) into its bidual. Moreover, the map j_p is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure μ on S is sigma-finite, then the dual of L¹(μ) is isometrically isomorphic to L^∞(μ) (more precisely, the map κ₁ corresponding to p = 1 is an isometry from L^∞(μ) onto L¹(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.

Embeddings

Colloquially, if 1 ≤ p < q ≤ ∞, L^p(S, μ) contains functions that are more locally singular, while elements of L^q(S, μ) 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:

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

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

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

Applications

L^p spaces are widely used in mathematics and applications.

Hausdorff-Young inequality

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

By contrast, if p > 2, the Fourier transform does not map into L^q.

Hilbert spaces

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L² and ℓ² are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ²(E), where E is a set with an appropriate cardinality.

Statistics

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

L^p for 0 < p < 1

Let (S, Σ, μ) be a measure space. If 0 < p < 1, then L^p(μ) can be defined as above: it is the vector space of those measurable functions f such that

$N_p(f) = \int_S |f|^p\, d\mu < \infty.$

As before, we may introduce the p-norm || f ||_p = N_p(f)^1/p, but || · ||_p does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality (a + b)^p ≤ a^p + b^p, valid for a ≥ 0 and b ≥ 0 implies that (Rudin 1991, §1.47)

$N_p(f+g)\le N_p(f) + N_p(g),$

and so the function

$d_p(f,g) = N_p(f-g) = \|f - g\|_p^p$

is a metric on L^p(μ). The resulting metric space is complete; the verification is similar to the familiar case when p ≥ 1.

In this setting L^p satisfies a reverse Minkowski inequality, that is for u and v in L^p

$\|\,|u|+|v|\,\|_p\geq \|u\|_p+\|v\|_p$ .

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces L^p for 1 < p < ∞ (Adams & Fournier 2003).

The space L^p for 0 < p < 1 is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is also locally bounded, much like the case p ≥ 1. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in ℓ^p or L^p([0, 1]), every open convex set containing the 0 function is unbounded for the p-quasi-norm; therefore, the 0 vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space S contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in L^p([0, 1]) is the entire space (Rudin 1991, §1.47). As a particular consequence, there are no nonzero linear functionals on L^p([0, 1]): the dual space is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space L^p(μ) = ℓ^p), the bounded linear functionals on ℓ^p are exactly those that are bounded on ℓ¹, namely those given by sequences in ℓ^∞. Although ℓ^p does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on Rⁿ, rather than work with L^p for 0 < p < 1, it is common to work with the Hardy space H^p whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the Hahn-Banach theorem still fails in H^p for p < 1 (Duren 1970, §7.5).

L⁰, the space of measurable functions

The vector space of (equivalence classes of) measurable functions on (S, Σ, μ) is denoted L⁰(S, Σ, μ) by many authors. It clearly contains all the L^p, and is equipped with the topology of convergence in measure (named convergence in probability in the special case of a probability measure μ, i.e. μ(S) = 1). The description is easier when μ is finite.

If μ is a finite measure on (S, Σ), the 0 function admits for the convergence in measure the following fundamental system of neighborhoods

$V_\varepsilon = \Bigl\{ f�: \mu \bigl(\{x�: |f(x)| > \varepsilon \} \bigr) < \varepsilon \Bigr\}, \ \ \varepsilon > 0.$

The topology can be defined by any metric d of the form

$d(f, g) = \int_S \varphi \bigl( |f(x) - g(x)| \bigr) \, \mathrm{d}\mu(x)$

where φ is bounded continuous concave and non-decreasing on [0, ∞), with φ(0) = 0 and φ(t) > 0 when t > 0 (for example, φ(t) = min(t, 1)). Such a metric is called Lévy-metric for L⁰. Under this metric the space L⁰ is complete (it is again an F-space). The space L⁰ is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure λ on Rⁿ, the definition of the fundamental system of neighborhoods could be modified as follows

$W_\varepsilon = \Bigl\{ f�: \lambda \bigl(\{ x�: |f(x)| > \varepsilon \ \text{and} \ |x| < 1 / \varepsilon\} \bigr) < \varepsilon \Bigr\}.$

The resulting space L⁰(Rⁿ, λ) coincides as topological vector space with L⁰(Rⁿ, g(x) dλ(x)), for any positive λ–integrable density g.

Weak L^p

Let (S, Σ, 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

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

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

$\lambda_f(t)\le \frac{\|f\|_p^p}{t^p}.$

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

$\lambda_f(t) \le \frac{C^p}{t^p}.$

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

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

The weak L^p coincide with the Lorentz spaces L^p,∞, so this notation is also used to denote them.

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

$\|f\|_{p,w}\le \|f\|_p,$

and in particular L^p(S, μ) ⊂ L^p,w(S, μ). Under the convention that two functions are equal if they are equal μ almost everywhere, then the spaces L^p,w are complete (Grafakos 2004).

For any 0 < r < p the expression

$||| f |||_{L^{p,\infty}}=\sup_{0<\mu(E)<\infty} \mu(E)^{-\frac{1}{r}+\frac{1}{p}}\left(\int_E |f|^r\,d\mu\right)^{\frac{1}{r}}$

is comparable to the L^p,w-norm. Further in the case p > 1, this expression defines a norm if r = 1. Hence for p > 1 the weak L^p spaces are Banach spaces (Grafakos 2004).

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

Weighted L^p spaces

As before, consider a measure space (S, Σ, μ). Let $w�: S \to [0, + \infty)$ be a measurable function. The w-weighted L^p space is defined as L^p(S, w dμ), where w dμ means the measure ν defined by

$\ \nu (A)�:= \int_{A} w(x) \, \mathrm{d} \mu (x), \ \ \ A \in \Sigma,$

or, in terms of the Radon-Nikodym derivative,

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

The norm for L^p(S, w dμ) is explicitly

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

As L^p-spaces, the weighted spaces have nothing special, since L^p(S, w dμ) is equal to L^p(S, ν). But they are the natural framework for several results in Harmonic Analysis; they appear for example in the Muckenhoupt theorem: for 1 < p < ∞, the classical Hilbert transform is defined on L^p(T, λ) where T denotes the unit circle and λ the Lebesgue measure; the (nonlinear) Hardy-Littlewood maximal operator is bounded on L^p(Rⁿ, λ). Muckenhoupt's theorem describes weights w such that the Hilbert transform remains bounded on L^p(T, w dλ) and the maximal operator on L^p(Rⁿ, w dλ).

References

Adams, Robert A. (1975), Sobolev Spaces, New York: Academic Press, ISBN 0-12-044150-0 .
Adams, Robert A.; Fournier, John F. (2003), Sobolev Spaces (Second ed.), Academic Press, ISBN 978-0120441433
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DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN 3-7643-4231-5 .
Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience .
Duren, P. (1970), Theory of H^p-Spaces, New York: Academic Press
Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253–257, ISBN 0-13-035399-X .
Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag .
Kalton, Nigel J.; Peck, N. Tenney; Roberts, James W. (1984), An F-space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge: Cambridge University Press, pp. xii+240, ISBN 0-521-27585-7 MR0808777
Riesz, Frigyes (1910), "Untersuchungen über Systeme integrierbarer Funktionen", Mathematische Annalen 69: 449-497
Rudin, Walter (1991), Functional Analysis, McGraw-Hill Science/Engineering/Math, ISBN 978-0-07-054236-5
Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, MR924157, ISBN 978-0-07-054234-1
Titchmarsh, EC (1976), The theory of functions, Oxford University Press, ISBN 9780198533498

External links

Proof that L^p spaces are complete on PlanetMath

Lp space

Contents

Motivation

When 0 < p < 1

ℓp spaces

Properties of ℓp spaces and the space c0

Lp spaces