In mathematics, the 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 , although according to the Bourbaki group they were first introduced by Frigyes Riesz .

spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.

Preliminaries

The p-norm in finite dimensions

-dimensional real vector space is given by the Euclidean norm

of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

For a real number the -norm or -norm is drawn from the set of real numbers, or one of its subsets.

-norm is the norm that corresponds to the rectilinear distance.

The -norm or maximum norm

-norms and maximum norm satisfy the properties of a "length function" (or norm), that is:

• only the zero vector has zero length,

• the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and

• the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).

-norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space.

Relations between p-norms

The grid distance or rectilinear distance (sometimes called the "Manhattan distance

:

of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.

This is a consequence of Hölder's inequality.

When 0 < p < 1

defines an absolutely homogeneous function however, the resulting function does not define a norm, because it is not subadditive defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-norm

defines a metric. The metric space

is a locally convex defined below, is no longer locally convex.

When p = 0

, no honest norm exists, and the replacement is an F-norm which is not locally convex

Sometimes a different "norm" is defined, but the topology it defines is not the usual one:. It gives the Hamming distance on the set.

ℓp spaces and sequence spaces

-norm can be extended to vectors that have an infinite number of components (sequences This contains as special cases:

• the space of sequences whose series are absolutely convergent,

• the space of square-summable sequences, which is a Hilbert space, and

• the space of bounded sequences.

The space of sequences has a natural vector space structure by applying scalar addition and multiplication. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex

Here, a complication arises, namely that the series-norm is finite.

(the harmonic series

-norm using the supremum of all bounded sequences. It turns out that

together with this norm is a Banach space.

General ℓp-space

over a general index set where convergence on the right requires that only countably many summands are nonzero (see also Absolute convergence over sets this is a non-separable Banach space which can be seen as the locally convex direct limit-sequence spaces.

-norm is even induced by a canonical inner product called the This inner product can be expressed in terms of the norm by using the polarization identity Define

can be turned into a measure space by giving it the discrete σ-algebra and the counting measure-space (defined below).

Lp spaces and Lebesgue integrals

-th power of the absolute value is Lebesgue integrable be a measure space of all measurable functions whose absolute value-th power has a finite integral, or in symbols:

are said to be , written (and its absolute value) is (or ) written is defined as the infimum then this is the same as the essential supremum:

almost everywhere

-norm.

-th power integrable functions

forms a vector space when addition and scalar multiplication are defined pointwise. although it is also a consequence of Minkowski's inequality satisfies the triangle inequality being absolutely homogeneous

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm defines a seminormed vector space. In general, the seminorm is not a norm but are not exists).

-seminorms

mentioned above.

almost everywhere have the same zero set

Quotient vector space

Like every seminorm induces a norm (defined shortly) on the canonical quotient vector space This normed quotient space is called and it is the subject of this article. We begin by defining the quotient vector space.

the coset almost everywhere consists of equivalence classes of functions.

-norm on the quotient vector space

on the coset on the quotient vector space This map is a norm called the ).

space

The normed vector space is called or the -th power integrable functions and it is a Banach space (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem

will be linearly isometrically isomorphic); in other words, they will be, up to a linear isometry space".

The above definitions generalize to Bochner spaces.

however, there is a theory of lifts enabling such recovery.

Special cases

are the natural numbers is the counting measure-norm as defined above.

is the only Hilbert space are sometimes called square-integrable functions, quadratically 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 .

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 defines a bounded operator space by multiplication.

When 0 < p < 1

does not satisfy the triangle inequality and defines only a quasi-norm The resulting metric space is complete, but not locally convex.

satisfies a reverse Minkowski inequality

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity .

is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex contains an infinite family of disjoint measurable sets of finite positive measure.

the continuous dual space is the zero space. In the case of the counting measure does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

it is common to work with the Hardy space 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 .

When p 0

For a finite-measurable functions, with functions identified if they are equal almost everywhere. The F-norm is defined by, the topology it defines is not locally convex.

with a weighted measure on the positive integers.

Properties

Hölder's inequality

and

This inequality, called Hölder's inequality where the supremum

Generalized Minkowski inequality

Minkowski inequality satisfies the triangle inequality

Atomic decomposition

has an , called are pairwise disjoint sets). being pairwise disjoint implies

denotes the indicator function causes no issues).

The complementary cumulative distribution function as the integral

Dual spaces

The dual space

is a well defined continuous linear mapping which is an isometry by the extremal case is a \sigma-finite measure space one can use the Radon–Nikodym theorem is an isometric isomorphism of Banach spaces. is the continuous dual space

is reflexive with the transpose

This map coincides with the canonical embedding is onto, as composition of two onto isometries, and this proves reflexivity.

is sigma-finite

can be identified with bounded signed finitely that are absolutely continuous See ba space 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

Embeddings

can be more spread out. Consider the Lebesgue measure need not decay at all but no blow-up is allowed. More formally:

1. does not contain sets of finite but arbitrarily large measure (e.g. any finite measure).

2. does not contain sets of non-zero but arbitrarily small measure (e.g. the counting measure).

Neither condition holds for the Lebesgue measure on the real line while both conditions holds for the counting measure on any finite set. As a consequence of the closed graph theorem, the embedding is continuous, i.e., the identity operator has finite measure, one can make the following explicit calculation using Hölder's inequality

The constant appearing in the above inequality is optimal, in the sense that the operator norm-almost-everywhere.

Dense subspaces

be a measure space and consider an integrable simple function is the indicator function By construction of the integral, the vector space of integrable simple functions is dense

is a normal topological space its Borel –algebra.

. Subsequently, there exists a Urysohn function

is the Lebesgue measure. For example, the space of continuous and compactly supported functions as well as the space of integrable step functions.

Closed subspaces

is a probability measure is finite-dimensional). In this theorem, which is due to Alexander Grothendieck, is Lebesgue measure on the unit circle

Applications

Statistics

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation metrics, and measures of central tendency can be characterized as solutions to variational problems.

In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the L^1 norm norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero). Elastic net regularization norm of the parameter vector.

Hausdorff–Young inequality

The Fourier transform for the real line (or, for periodic functions, see Fourier series This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.

Hilbert spaces

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus

Generalizations and extensions

Weak Lp

a measurable function The distribution function

then by Markov's inequality

is said to be in the space

coincide with the Lorentz spaces so this notation is also used to denote them.

-norm is not a true norm, since the triangle inequality

are complete .

spaces are Banach spaces .

-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.

Weighted Lp spaces

As before, consider a measure space- space

or, in terms of the Radon–Nikodym derivative the norm

But they are the natural framework for several results in harmonic analysis ; they appear for example in the Muckenhoupt theorem the classical Hilbert transform denotes the unit circle the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator

Lp spaces on manifolds

on a manifold, called the spaces of the manifold, using densities.

Vector-valued Lp spaces

and a locally convex space (here assumed to be complete in a number of ways. One way is to define the spaces of Bochner integrable and Pettis integrable functions, and then endow them with locally convex TVS-topologies topology. Another way involves topological tensor products This tensor product is then endowed with a locally convex topology that turns it into a topological tensor product, the most common of which are the projective tensor product and the injective tensor product In general, neither of these space are complete so their completions (this is analogous to how the space of scalar-valued simple functions). Alexander Grothendieck is a nuclear space (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.

L0 space of measurable functions

The vector space of (equivalence classes and is equipped with the topology of convergence in measure), this mode of convergence is named convergence in probability is always a topological abelian group but is only a topological vector space-finite then the weaker topology of local convergence in measure is an F-space, i.e. a completely metrizable topological vector space for a suitable choice of probability measure

is a finite measure function admits for the convergence in measure the following fundamental system of neighborhoods

Such a metric is called Lévy is in general not locally bounded, and not locally convex.

See also

Notes

References

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External links

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