# L^{p} space

In mathematics, the **L ^{p} spaces** are function spaces defined using a natural generalization of the

*p*-norm for finite-dimensional vector spaces. They are sometimes called

**Lebesgue spaces**, named after Henri Lebesgue Template:Harv, although according to the Bourbaki group Template:Harv they were first introduced by Frigyes Riesz Template:Harv.

**L**form an important class of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.

^{p}spaces## Contents

- 1 The Template:Mvar-norm in finite dimensions
- 2 The Template:Mvar-norm in countably infinite dimensions
- 3
*L*spaces^{p} - 4 Properties of
*L*^{p}spaces - 5 Applications
- 6
*L*(0 <^{p}*p*< 1) - 7 Weak
*L*^{p} - 8 Weighted
*L*spaces^{p} - 9
*L*spaces on manifolds^{p} - 10 See also
- 11 Notes
- 12 References
- 13 External links

## The Template:Mvar-norm in finite dimensions

The length of a vector *x* = (*x*_{1}, *x*_{2}, ..., *x _{n}*) in the Template:Mvar-dimensional real vector space

**R**

^{n}is usually given by the Euclidean norm:

The Euclidean distance between two points Template:Mvar and Template:Mvar is the length Template:!!*x* − *y*Template:!!_{2} of the straight line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. An analogy to this can be found in Manhattan taxi drivers who should measure distance not in terms of the length of the straight line to their destination, but in terms of the Manhattan distance, which takes into account that streets are either orthogonal or parallel to each other. The class of Template:Mvar-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

### Definition

For a real number *p* ≥ 1, the **Template:Mvar-norm** or ** L^{p}-norm** of Template:Mvar is defined by

(The absolute value bars are unnecessary if *p* is a rational number with even numerator and odd denominator.)

The Euclidean norm from above falls into this class and is the 2-norm, and the 1-norm is the norm that corresponds to the Manhattan distance.

The ** L^{∞}-norm** or maximum norm (or uniform norm) is the limit of the

*L*-norms for

^{p}*p*→ ∞. It turns out that this limit is equivalent to the following definition:

For all *p* ≥ 1, the Template:Mvar-norms and maximum norm as defined above indeed satisfy 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).

Abstractly speaking, this means that **R**^{n} together with the Template:Mvar-norm is a Banach space. This Banach space is the ** L^{p}-space** over

**R**

^{n}.

#### Relations between Template:Mvar-norms

The grid distance ("Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:

This fact generalizes to Template:Mvar-norms in that the Template:Mvar-norm Template:!!*x*Template:!!_{p} of any given vector Template:Mvar does not grow with Template:Mvar:

- Template:!!
*x*Template:!!_{p+a}≤ Template:!!*x*Template:!!_{p}for any vector Template:Mvar and real numbers*p*≥ 1 and*a*≥ 0. (In fact this remains true for 0 <*p*< 1 and*a*≥ 0.)

For the opposite direction, the following relation between the 1-norm and the 2-norm is known:

This inequality depends on the dimension Template:Mvar of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.

In general, for vectors in **C**^{n} where 0 < *r* < *p*:

### When 0 < *p* < 1

In **R**^{n} for *n* > 1, the formula

defines an absolutely homogeneous function of degree 1 for 0 < *p* < 1; however, the resulting function does not define an F-norm, because it is not subadditive. In **R**^{n} for *n* > 1, the formula for 0 < *p* < 1

defines a subadditive function, which does define an F-norm. This F-norm is homogeneous of degree Template:Mvar.

However, the function

defines a metric. The metric space (**R**^{n}, *d*_{p}) is denoted by ℓ_{n}^{p}.

Although the Template:Mvar-unit ball *B*_{n}^{p} around the origin in this metric is "concave", the topology defined on **R**^{n} by the metric *d _{p}* is the usual vector space topology of

**R**

^{n}, 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 Template:Mvar such that the multiple

*C*

*B*

_{n}

^{p}of the Template:Mvar-unit ball contains the convex hull of

*B*

_{n}

^{p}, equal to

*B*

_{n}

^{1}. The fact that for fixed

*p*< 1 we have

shows that the infinite-dimensional sequence space *ℓ ^{p}* defined below, is no longer locally convex.

### When *p* = 0

There is one ℓ_{0} norm and another function called the ℓ_{0} "norm" (with quotation marks).

The mathematical definition of the ℓ_{0} norm was established by Banach's *Theory of Linear Operations*. The space of sequences has a complete metric topology provided by the F-norm

which is discussed by Stefan Rolewicz in *Metric Linear Spaces*.^{[1]} The ℓ_{0}-normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the ℓ_{0} "norm" by David Donoho — whose quotation marks warn that this function is not a proper norm — is the number of non-zero entries of the vector *x*. Many authors abuse terminology by omitting the quotation marks. Defining 0^{0} = 0, the zero "norm" of *x* is equal to

This is not a norm (B-norm, with "B" for Banach) because it is not homogeneous. Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics – notably in compressed sensing in signal processing and computational harmonic analysis.

## The Template:Mvar-norm in countably infinite dimensions

Template:Rellink
The Template:Mvar-norm can be extended to vectors that have an infinite number of components, which yields the space *ℓ ^{ p}*. This contains as special cases:

*ℓ*^{ 1}, the space of sequences whose series is absolutely convergent,*ℓ*^{ 2}, 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, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by:

Define the Template:Mvar-norm:

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 Template:Mvar-norm for 1 ≤ *p* < ∞. The space *ℓ ^{ p}* is then defined as the set of all infinite sequences of real (or complex) numbers such that the Template:Mvar-norm is finite.

One can check that as Template:Mvar increases, the set *ℓ ^{ p}* grows larger. For example, the sequence

is not in *ℓ*^{ 1}, but it is in *ℓ ^{ p}* for

*p*> 1, as the series

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

One also defines the ∞-norm using the supremum:

and the corresponding space *ℓ*^{ ∞} of all bounded sequences. It turns out that^{[2]}

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 Template:Mvar-norm thus defined on *ℓ ^{ p}* is indeed a norm, and

*ℓ*together with this norm is a Banach space. The fully general

^{ p}*L*space is obtained — as seen below — by considering vectors, not only with finitely or countably-infinitely many components, but with "

^{p}*arbitrarily many components*"; in other words, functions. An integral instead of a sum is used to define the Template:Mvar-norm.

*L*^{p} spaces

^{p}

An *L ^{p}* space may be defined as a space of functions for which the

*p*-th power of the absolute value is Lebesgue integrable.

^{[3]}More generally, let 1 ≤

*p*< ∞ and (

*S*, Σ,

*μ*) be a measure space. Consider the set of all measurable functions from Template:Mvar to

**C**or

**R**whose absolute value raised to the Template:Mvar-th power has finite integral, or equivalently, that

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

for every scalar Template:Mvar.

That the sum of two Template:Mvar-th power integrable functions is again Template:Mvar-th power integrable follows from the inequality

(This comes from the convexity of for .)

In fact, more is true. Minkowski's inequality says the triangle inequality holds for Template:!! · Template:!!_{p}. Thus the set of Template:Mvar-th power integrable functions, together with the function Template:!! · Template:!!_{p}, is a seminormed vector space, which is denoted by .

{{safesubst:#invoke:anchor|main}}
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 Template:!! · Template:!!_{p}. Since for any measurable function *f* , we have that Template:!! *f* Template:!!_{p} = 0 if and only if *f* = 0 almost everywhere, the kernel of Template:!! · Template:!!_{p} does not depend upon Template:Mvar,

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

For *p* = ∞, the space *L*^{∞}(*S*, *μ*) is defined as follows. We start with the set of all measurable functions from Template:Mvar 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 a function *f* in this set, its essential supremum serves as an appropriate norm:

As before, if there exists *q* < ∞ such that *f* ∈ *L*^{∞}(*S*, *μ*) ∩ *L ^{q}*(

*S*,

*μ*), then

For 1 ≤ *p* ≤ ∞, *L ^{p}*(

*S*,

*μ*) is a Banach space. The fact that

*L*is complete is often referred to as the Riesz-Fischer theorem. Completeness can be checked using the convergence theorems for Lebesgue integrals.

^{p}When the underlying measure space Template:Mvar is understood, *L ^{p}*(

*S*,

*μ*) is often abbreviated

*L*(

^{p}*μ*), or just

*L*. The above definitions generalize to Bochner spaces.

^{p}### Special cases

Similar to the ℓ^{p} spaces, *L*^{2} is the only Hilbert space among *L ^{p}* spaces. In the complex case, the inner product on

*L*

^{2}is defined by

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in *L*^{2} are sometimes called **quadratically integrable functions**, **square-integrable functions** or **square-summable functions**, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral Template:Harv.

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.

For 1 ≤ *p* ≤ ∞ the ℓ^{p} spaces are a special case of *L ^{p}* spaces, when

*S*=

**N**, and Template:Mvar is the counting measure on

**N**. More generally, if one considers any set Template:Mvar with the counting measure, the resulting

*L*space is denoted ℓ

^{p}^{p}(

*S*). For example, the space ℓ

^{p}(

**Z**) is the space of all sequences indexed by the integers, and when defining the Template:Mvar-norm on such a space, one sums over all the integers. The space ℓ

^{p}(

*n*), where Template:Mvar is the set with Template:Mvar elements, is

**R**

^{n}with its Template:Mvar-norm as defined above. As any Hilbert space, every space

*L*

^{2}is linearly isometric to a suitable ℓ

^{2}(

*I*), where the cardinality of the set Template:Mvar is the cardinality of an arbitrary Hilbertian basis for this particular

*L*

^{2}.

## 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 Template:Mvar is such that {{ safesubst:#invoke:Unsubst||$B=1/

*p*}} + {{ safesubst:#invoke:Unsubst||$B=1/

*q*}} = 1. This isomorphism associates

*g*∈

*L*(

^{q}*μ*) with the functional

*κ*(

_{p}*g*) ∈

*L*(

^{p}*μ*)

^{∗}defined by

The fact that *κ _{p}*(

*g*) is well defined and continuous follows from Hölder's inequality.

*κ*:

_{p}*L*(

^{q}*μ*) →

*L*(

^{p}*μ*)

^{∗}is a linear mapping 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, see

^{[4]}) that any

*G*∈

*L*(

^{p}*μ*)

^{∗}can be expressed this way: i.e., that

*κ*is

_{p}*onto*. Since

*κ*is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say simply that

_{p}*L*"

^{q}*is*" the dual of

*L*.

^{p}For 1 < *p* < ∞, the space *L ^{p}*(

*μ*) is reflexive. Let

*κ*be as above and let

_{p}*κ*:

_{q}*L*(

^{p}*μ*) →

*L*(

^{q}*μ*)

^{∗}be the corresponding linear isometry. Consider the map from

*L*(

^{p}*μ*) to

*L*(

^{p}*μ*)

^{∗∗}, obtained by composing

*κ*with the transpose (or adjoint) of the inverse of

_{q}*κ*:

_{p}This map coincides with the canonical embedding Template:Mvar of *L ^{p}*(

*μ*) into its bidual. Moreover, the map

*j*is onto, as composition of two onto isometries, and this proves reflexivity.

_{p}If the measure Template:Mvar on Template:Mvar is sigma-finite, then the dual of *L*^{1}(*μ*) is isometrically isomorphic to *L*^{∞}(*μ*) (more precisely, the map *κ*_{1} corresponding to *p* = 1 is an isometry from *L*^{∞}(*μ*) onto *L*^{1}(*μ*)^{∗}).

The dual of *L*^{∞} is subtler. Elements of *L*^{∞}(*μ*)^{∗} can be identified with bounded signed *finitely* additive measures on Template:Mvar that are absolutely continuous with respect to Template:Mvar. See ba space for more details. If we assume the axiom of choice, this space is much bigger than *L*^{1}(*μ*) except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo-Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of *ℓ*^{∞} is *ℓ*^{1}.^{[5]}

### Embeddings

Colloquially, if 1 ≤ *p* < *q* ≤ ∞, then *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

*L*

^{1}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:

^{[6]}

- Let 0 ≤
*p*<*q*≤ ∞.*L*(^{q}*S*,*μ*) ⊂*L*(^{p}*S*,*μ*) iff Template:Mvar does not contain sets of arbitrarily large measure, and - Let 0 ≤
*p*<*q*≤ ∞.*L*(^{p}*S*,*μ*) ⊂*L*(^{q}*S*,*μ*) iff Template:Mvar does not contain arbitrarily small sets of non-zero measure.

In both cases the embedding is continuous, in that the identity operator is a bounded linear map from
*L ^{q}* to

*L*in the first case, and

^{p}*L*to

^{p}*L*in the second. (This is a consequence of the closed graph theorem and properties of

^{q}*L*spaces.) Indeed, if the domain Template:Mvar has finite measure, one can make the following explicit calculation via Jensen's inequality:

^{p}The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity *I* : *L ^{q}*(

*S*,

*μ*) →

*L*(

^{p}*S*,

*μ*) is precisely

the case of equality being achieved exactly when *f* = 1 Template:Mvar-a.e.

### Dense subspaces

Throughout this section we assume that: 1 ≤ *p* < ∞.

Let (*S*, Σ, *μ*) be a measure space. An *integrable simple function* *f* on Template:Mvar is one of the form

where *a _{j}* is scalar,

*A*∈ Σ has finite measure and is the indicator function of the set , for

_{j}*j*= 1, ...,

*n*. By construction of the integral, the vector space of integrable simple functions is dense in

*L*(

^{p}*S*, Σ,

*μ*).

More can be said when Template:Mvar is a metrizable topological space and Σ its [[Borel algebra|Borel Template:Mvar–algebra]], i.e., the smallest Template:Mvar–algebra of subsets of Template:Mvar containing the open sets.

Suppose *V* ⊂ *S* is an open set with *μ*(*V*) < ∞. It can be proved that for every Borel set *A* ∈ Σ contained in Template:Mvar, and for every *ε* > 0, there exist a closed set Template:Mvar and an open set Template:Mvar such that

It follows that there exists Template:Mvar continuous on Template:Mvar such that

If Template:Mvar can be covered by an increasing sequence (*V _{n}*) of open sets that have finite measure, then the space of Template:Mvar–integrable continuous functions is dense in

*L*(

^{p}*S*, Σ,

*μ*). More precisely, one can use bounded continuous functions that vanish outside one of the open sets

*V*.

_{n}This applies in particular when *S* = **R**^{d} and when Template:Mvar is the Lebesgue measure. The space of continuous and compactly supported functions is dense in *L ^{p}*(

**R**

^{d}). Similarly, the space of integrable

*step functions*is dense in

*L*(

^{p}**R**

^{d}); this space is the linear span of indicator functions of bounded intervals when

*d*= 1, of bounded rectangles when

*d*= 2 and more generally of products of bounded intervals.

Several properties of general functions in *L ^{p}*(

**R**

^{d}) are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on

*L*(

^{p}**R**

^{d}), in the following sense:

where

## 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, see 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 *L*^{2} and ℓ^{2} are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ^{2}(*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} (0 < *p* < 1)

^{p}

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

As before, we may introduce the Template:Mvar-norm Template:!! *f* Template:!!_{p} = *N _{p}*(

*f*)

^{1/p}, but Template:!! · Template:!!

_{ p}does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality (

*a*+

*b*)

^{ p}≤

*a*+

^{ p}*b*, valid for

^{ p}*a*,

*b*≥ 0 implies that Template:Harv

and so the function

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*,

*v*in

*L*

^{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*< ∞ Template:Harv.

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*or

^{ p}*L*([0, 1]), every open convex set containing the 0 function is unbounded for the Template:Mvar-quasi-norm; therefore, the 0 vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space Template:Mvar contains an infinite family of disjoint measurable sets of finite positive measure.

^{p}The only nonempty convex open set in *L ^{p}*([0, 1]) is the entire space Template:Harv. As a particular consequence, there are no nonzero linear functionals on

*L*([0, 1]): the dual space is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space

^{p}*L*(

^{p}*μ*) =

*ℓ*), the bounded linear functionals on

^{ p}*ℓ*are exactly those that are bounded on

^{ p}*ℓ*

^{ 1}, namely those given by sequences in

*ℓ*

^{ ∞}. Although

*ℓ*does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

^{ p}The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on **R**^{n}, rather than work with *L ^{p}* for 0 <

*p*< 1, it is common to work with the Hardy space

*H*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

^{ p}*H*for

^{ p}*p*< 1 Template:Harv.

*L*^{0}, the space of measurable functions

The vector space of (equivalence classes of) measurable functions on (*S*, Σ, *μ*) is denoted *L*^{0}(*S*, Σ, *μ*) Template:Harv. By definition, it contains all the *L ^{p}*, and is equipped with the topology of

*convergence in measure*. When Template:Mvar is a probability measure (i.e.,

*μ*(

*S*) = 1), this mode of convergence is named

*convergence in probability*.

The description is easier when Template:Mvar is finite. If Template:Mvar is a finite measure on (*S*, Σ), the 0 function admits for the convergence in measure the following fundamental system of neighborhoods

The topology can be defined by any metric Template:Mvar of the form

where Template:Mvar 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 *L*^{0}. Under this metric the space *L*^{0} is complete (it is again an F-space). The space *L*^{0} is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure Template:Mvar on **R**^{n}, the definition of the fundamental system of neighborhoods could be modified as follows

The resulting space *L*^{0}(**R**^{n}, *λ*) coincides as topological vector space with *L*^{0}(**R**^{n}, *g*(*x*) d*λ*(x)), for any positive Template:Mvar–integrable density Template:Mvar.

## Weak *L*^{p}

^{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

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

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,

The best constant *C* for this inequality is the *L ^{p,w}*-norm of

*f*, and is denoted by

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*),

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 Template:Harv.

For any 0 < *r* < *p* the expression

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 Template:Harv.

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

^{p}

As before, consider a measure space (*S*, Σ, *μ*). Let *w* : *S* → [0, ∞) be a measurable function. The Template:Mvar-**weighted L^{p} space** is defined as

*L*(

^{p}*S*,

*w*d

*μ*), where

*w*d

*μ*means the measure Template:Mvar defined by

or, in terms of the Radon–Nikodym derivative, *w* = {{ safesubst:#invoke:Unsubst||$B=d*ν*/d*μ*}} the norm for *L ^{p}*(

*S*,

*w*d

*μ*) is explicitly

As *L ^{p}*-spaces, the weighted spaces have nothing special, since

*L*(

^{p}*S*,

*w*d

*μ*) is equal to

*L*(

^{p}*S*, d

*ν*). But they are the natural framework for several results in harmonic analysis Template:Harv; 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 Template:Mvar the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on

*L*(

^{p}**R**

^{n},

*λ*). Muckenhoupt's theorem describes weights Template:Mvar such that the Hilbert transform remains bounded on

*L*(

^{p}**T**,

*w*d

*λ*) and the maximal operator on

*L*(

^{p}**R**

^{n},

*w*d

*λ*).

*L*^{p} spaces on manifolds

^{p}

One may also define spaces *L ^{p}*(

*M*) on a manifold, called the

**intrinsic**of the manifold, using densities.

*L*spaces^{p}## See also

- Birnbaum–Orlicz space
- Hardy space
- Riesz–Thorin theorem
- Hölder mean
- Hölder space
- Root mean square
- Locally integrable function
- spaces over a locally compact group
- Minkowski distance

## Notes

- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}, page 16
- ↑ We could just say "integrable". Since the integrand is a non-negative real-valued function, there is no difference between having a finite Lebesgue integral and having a finite improper integral (as there is say for the function sin(
*x*)/*x*when integrated over the entire real line). - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}, Theorem 6.16
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }} See Sections 14.77 and 27.44--47
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}

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## External links

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