In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the spaces are
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s defined using a natural generalization of the
-norm for finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s. They are sometimes called Lebesgue spaces, named after
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
, although according to the
Bourbaki group they were first introduced by
Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...
. spaces form an important class of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, and of
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s. 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.
Applications
Statistics
In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, measures of
central tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...
and
statistical dispersion
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a Probability distribution, distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard de ...
, such as the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
,
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
, and
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
, are defined in terms of metrics, and measures of central tendency can be characterized as
solutions to variational problems.
In
penalized regression
Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution.
RLS is used for two main reasons. The first comes up when the number of variables ...
, "L1 penalty" and "L2 penalty" refer to penalizing either the
norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its norm (its
Euclidean length
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
). Techniques which use an L1 penalty, like
LASSO
A lasso ( or ), also called lariat, riata, or reata (all from Castilian, la reata 're-tied rope'), is a loop of rope designed as a restraint to be thrown around a target and tightened when pulled. It is a well-known tool of the Spanish an ...
, encourage solutions where many parameters are zero. Techniques which use an L2 penalty, like
ridge regression
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also ...
, encourage solutions where most parameter values are small.
Elastic net regularization
In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods.
Specification
The elas ...
uses a penalty term that is a combination of the norm and the norm of the parameter vector.
Hausdorff–Young inequality
The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
for the real line (or, for
periodic functions
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
, see
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
), maps to (or to ) respectively, where and This is a consequence of the
Riesz–Thorin interpolation theorem, and is made precise with the
Hausdorff–Young inequality
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by and extended by . It is now typically understood as a rather direct corollary of th ...
.
By contrast, if , the Fourier transform does not map into .
Hilbert spaces
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s are central to many applications, from
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
to
stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
. The spaces and are both Hilbert spaces. In fact, by choosing a Hilbert basis , i.e., a maximal orthonormal subset of or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to (same as above), i.e., a Hilbert space of type .
The -norm in finite dimensions
The length of a vector in the -dimensional
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
is usually given by the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
:
The Euclidean distance between two points and is the length 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 is suggested by 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
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...
, which takes into account that streets are either orthogonal or parallel to each other. The class of -norms generalizes these two examples and has an abundance of applications in many parts of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
.
Definition
For a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, the -norm or -norm of is defined by
The absolute value bars can be dropped when is a rational number with an even numerator in its reduced form, and is drawn from the set of real numbers, or one of its subsets.
The Euclidean norm from above falls into this class and is the -norm, and the -norm is the norm that corresponds to the
rectilinear distance
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...
.
The -norm or
maximum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when the ...
(or uniform norm) is the limit of the -norms for . It turns out that this limit is equivalent to the following definition:
See
-infinity.
For all , the -norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
), 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 (
positive homogeneity
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
), and
*the length of the sum of two vectors is no larger than the sum of lengths of the vectors (
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
).
Abstractly speaking, this means that together with the -norm is a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
. Moreover, it turns out that this space is complete, thus making it a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. This Banach space is the -space over .
Relations between -norms
The grid distance or rectilinear distance (sometimes called the "
Manhattan distance
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences ...
") 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 -norms in that the -norm of any given vector does not grow with :
For the opposite direction, the following relation between the -norm and the -norm is known:
This inequality depends on the dimension of the underlying vector space and follows directly from the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
.
In general, for vectors in where :
This is a consequence of
Hölder's inequality.
When
In for , the formula
defines an absolutely
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
for ; however, the resulting function does not define a norm, because it is not
subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
. On the other hand, the formula
defines a subadditive function at the cost of losing absolute homogeneity. It does define an
F-norm, though, which is homogeneous of degree .
Hence, the function
defines a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
. The metric space is denoted by .
Although the -unit ball around the origin in this metric is "concave", the topology defined on by the metric is the usual vector space topology of , hence is a
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of is to denote by the smallest constant such that the multiple of the -unit ball contains the convex hull of , equal to . The fact that for fixed we have
shows that the infinite-dimensional sequence space defined below, is no longer locally convex.
When
There is one norm and another function called the "norm" (with quotation marks).
The mathematical definition of the norm was established by
Banach's ''
Theory of Linear Operations
A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or resear ...
''. The
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
of sequences has a complete metric topology provided by the
F-norm
which is discussed by Stefan Rolewicz in ''Metric Linear Spaces''.
The -normed space is studied in functional analysis, probability theory, and harmonic analysis.
Another function was called the "norm" by
David Donoho
David Leigh Donoho (born March 5, 1957) is an American statistician. He is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences. His work includes the develop ...
—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector . Many authors
abuse terminology by omitting the quotation marks. Defining
, the zero "norm" of is equal to
![Lp space animation](https://upload.wikimedia.org/wikipedia/commons/1/1f/Lp_space_animation.gif)
This is not a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
because it is not
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
. For example, scaling the vector by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in
scientific computing
Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...
,
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
, and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
–notably in
compressed sensing
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a Signal (electronics), signal, by finding solutions to Underdetermined ...
in
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
and computational
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
. Despite not being a norm, the associated metric, known as
Hamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
, is a valid distance, since homogeneity is not required for distances.
The -norm in infinite dimensions and spaces
The sequence space
The -norm can be extended to vectors that have an infinite number of components (
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s), which yields the space . This contains as special cases:
*, the space of sequences whose series is
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
,
*, the space of square-summable sequences, which is a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, and
*, the space of
bounded sequence
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that
:, f(x), \le M
for all ''x'' in ''X''. A ...
s.
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
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of real (or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
) numbers are given by:
Define the -norm:
Here, a complication arises, namely that the
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
on the right is not always convergent, so for example, the sequence made up of only ones, , will have an infinite -norm for . The space is then defined as the set of all infinite sequences of real (or complex) numbers such that the -norm is finite.
One can check that as increases, the set grows larger. For example, the sequence
is not in , but it is in for , as the series
diverges for (the
harmonic series), but is convergent for .
One also defines the -norm using the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
:
and the corresponding space of all bounded sequences. It turns out that
if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider spaces for .
The -norm thus defined on is indeed a norm, and together with this norm is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. The fully general 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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
instead of a sum is used to define the -norm.
General ℓ''p''-space
In complete analogy to the preceding definition one can define the space
over a general
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
(and
) as
where convergence on the right means that only countably many summands are nonzero (see also
Unconditional convergence In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not al ...
).
With the norm
the space
becomes a Banach space.
In the case where
is finite with
elements, this construction yields with the
-norm defined above.
If
is countably infinite, this is exactly the sequence space
defined above.
For uncountable sets
this is a non-
separable Banach space which can be seen as the
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of
-sequence spaces.
For
the
-norm is even induced by a canonical
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
called the ', which means that
holds for all vectors
This inner product can expressed in terms of the norm by using the
polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product then t ...
.
On
it can be defined by
while for the space
associated with a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
which consists of all
square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
s, it is
Now consider the case
We can define
where for all ''x''
The index set
can be turned into a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
by giving it the
discrete σ-algebra and the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...
. Then the space
is just a special case of the more general
-space (see below).
''Lp'' spaces and Lebesgue integrals
An space may be defined as a space of measurable functions for which the
-th power of the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
is
Lebesgue integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
, where functions which agree almost everywhere are identified. More generally, let and be a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
. Consider the set of all
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
s from to or whose
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
raised to the -th power has a finite integral, or equivalently, that
The set of such functions forms a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, with the following natural operations:
for every scalar .
That the sum of two -th power integrable functions is again -th power integrable follows from the inequality
(This comes from the convexity of
for
.)
In fact, more is true. ''
Minkowski's inequality
In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the tr ...
'' says the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
holds for . Thus the set of -th power integrable functions, together with the function , is a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
ed vector space, which is denoted by
.
For , the space
is the space of measurable functions bounded almost everywhere, with (when μ(X)≠0) the
essential supremum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
of its absolute value as a norm:
As in the discrete case, if there exists such that , then
can be made into a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
in a standard way; one simply takes the
quotient space with respect to the subspace of functions whose p-norm is zero. Since for any measurable function , we have that if and only if
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, that subspace does not depend upon ,
In the quotient space, two functions and are identified if almost everywhere. The resulting normed vector space is, by definition,
In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of
in
. For
, however, there is a
theory of lifts enabling such recovery.
When the underlying measure space is understood, is often abbreviated , or just .
For is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. The fact that is complete is often referred to as the
Riesz-Fischer theorem, and can be proven using the convergence theorems for
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
s.
The above definitions generalize to
Bochner space
In mathematics, Bochner spaces are a generalization of the concept of L^p spaces to functions whose values lie in a Banach space which is not necessarily the space \R or \Complex of real or complex numbers.
The space L^p(X) consists of (equival ...
s.
Special cases
Similar to the spaces, is the only
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
among spaces. In the complex case, the inner product on is defined by
The additional inner product structure allows for a richer theory, with applications to, for instance,
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Functions in are sometimes called
square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
s, 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
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
.
If we use complex-valued functions, the space is a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative
von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algeb ...
. An element of defines a
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
on any space by
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
.
For the spaces are a special case of spaces, when , and is the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...
on . More generally, if one considers any set with the counting measure, the resulting space is denoted . For example, the space is the space of all sequences indexed by the integers, and when defining the -norm on such a space, one sums over all the integers. The space , where is the set with elements, is with its -norm as defined above. As any Hilbert space, every space is linearly isometric to a suitable , where the cardinality of the set is the cardinality of an arbitrary Hilbertian basis for this particular .
Properties of ''L''''p'' spaces
Dual spaces
The
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
(the Banach space of all continuous linear functionals) of for has a natural isomorphism with , where is such that (i.e. ). This isomorphism associates with the functional defined by
for every
The fact that is well defined and continuous follows from
Hölder's inequality. is a linear mapping which is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
by the
extremal case of Hölder's inequality. It is also possible to show (for example with the
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
, see) that any can be expressed this way: i.e., that is ''onto''. Since is onto and isometric, it is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s. With this (isometric) isomorphism in mind, it is usual to say simply that is the dual Banach space of .
For , the space is
reflexive. Let be as above and let be the corresponding linear isometry. Consider the map from to , obtained by composing with the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
(or adjoint) of the inverse of :
This map coincides with the
canonical embedding of into its bidual. Moreover, the map is onto, as composition of two onto isometries, and this proves reflexivity.
If the measure on is
sigma-finite, then the dual of is isometrically isomorphic to (more precisely, the map corresponding to is an isometry from onto ).
The dual of is subtler. Elements of can be identified with bounded signed ''finitely'' additive measures on that are
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
with respect to . See
ba space
In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma. The norm is defined as the variation, that is \, \nu\, =, \nu, (X).
If Σ is ...
for more details. If we assume the axiom of choice, this space is much bigger than except in some trivial cases. However,
Saharon Shelah
Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.
Biography
Shelah was born in Jerusalem on July 3, ...
proved that there are relatively consistent extensions of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(ZF +
DC + "Every subset of the real numbers has the
Baire property
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such t ...
") in which the dual of is .
[ See Sections 14.77 and 27.44–47]
Embeddings
Colloquially, if , then contains functions that are more locally singular, while elements of can be more spread out. Consider the Lebesgue measure on the half line . A continuous function in might blow up near but must decay sufficiently fast toward infinity. On the other hand, continuous functions in need not decay at all but no blow-up is allowed. The precise technical result is the following.
Suppose that . Then:
# if and only if does not contain sets of finite but arbitrarily large measure, and
# if and only if does not contain sets of non-zero but arbitrarily small measure.
Neither condition holds for the real line with the Lebesgue measure. In both cases the embedding is continuous, in that the identity operator is a bounded linear map from
to in the first case,
and to in the second.
(This is a consequence of the
closed graph theorem
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
Each gives conditions when functions with closed graphs are necessarily continuous.
Graphs and m ...
and properties of spaces.) Indeed, if the domain has finite measure,
one can make the following explicit calculation using
Hölder's inequality
leading to
The constant appearing in the above inequality is optimal, in the sense that the
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
of the identity is precisely
the case of equality being achieved exactly when -almost-everywhere.
Dense subspaces
Throughout this section we assume that: .
Let be a measure space. An ''integrable simple function'' on is one of the form
where is scalar, has finite measure and
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of the set
, for . By construction of the
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
, the vector space of integrable simple functions is dense in .
More can be said when is a
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
and its
Borel –algebra, i.e., the smallest –algebra of subsets of containing the
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s.
Suppose is an open set with . It can be proved that for every Borel set contained in , and for every , there exist a closed set and an open set such that
It follows that there exists a continuous
Urysohn function on that is on and on , with
If can be covered by an increasing sequence of open sets that have finite measure, then the space of –integrable continuous functions is dense in . More precisely, one can use bounded continuous functions that vanish outside one of the open sets .
This applies in particular when and when is the Lebesgue measure. The space of continuous and compactly supported functions is dense in . Similarly, the space of integrable ''step functions'' is dense in ; this space is the linear span of indicator functions of bounded intervals when , of bounded rectangles when and more generally of products of bounded intervals.
Several properties of general functions in 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 , in the following sense:
where
Let be a measure space. If , then can be defined as above: it is the vector space of those measurable functions such that
As before, we may introduce the -norm , but does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality , valid for implies that
and so the function
is a metric on . The resulting metric space is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
; the verification is similar to the familiar case when .
In this setting satisfies a ''reverse Minkowski inequality'', that is for in
This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniformly convex space, uniform convexity of the spaces for .
The space for 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 . It is the prototypical example of an F-space that, for most reasonable measure spaces, is not
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
: in or , every open convex set containing the function is unbounded for the -quasi-norm; therefore, the vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space contains an infinite family of disjoint measurable sets of finite positive measure.
The only nonempty convex open set in is the entire space . As a particular consequence, there are no nonzero linear functionals on : the dual space is the zero space. In the case of the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...
on the natural numbers (producing the sequence space ), the bounded linear functionals on are exactly those that are bounded on , namely those given by sequences in . Although 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 , rather than work with for , 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 for .
, the space of measurable functions
The vector space of (equivalence classes of) measurable functions on is denoted . By definition, it contains all the , and is equipped with the topology of ''convergence in measure''. When is a probability measure (i.e., ), this mode of convergence is named ''convergence in probability''.
The description is easier when is finite. If is a finite measure on , the function admits for the convergence in measure the following fundamental system of neighborhoods
The topology can be defined by any metric of the form
where is bounded continuous concave and non-decreasing on , with and when (for example, . Such a metric is called Paul Lévy (mathematician), Lévy-metric for . Under this metric the space is complete (it is again an F-space). The space is in general not locally bounded, and not locally convex.
For the infinite Lebesgue measure on , the definition of the fundamental system of neighborhoods could be modified as follows
The resulting space coincides as topological vector space with , for any positive –integrable density .
Generalizations and extensions
Weak
Let be a measure space, and a
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
with real or complex values on . The cumulative distribution function, distribution function of is defined for by
If is in for some with , then by Markov's inequality,
A function is said to be in the space weak , or , if there is a constant such that, for all ,
The best constant for this inequality is the -norm of , and is denoted by
The weak coincide with the Lorentz spaces , so this notation is also used to denote them.
The -norm is not a true norm, since the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
fails to hold. Nevertheless, for in ,
and in particular .
In fact, one has
and raising to power and taking the supremum in one has
Under the convention that two functions are equal if they are equal almost everywhere, then the spaces are complete .
For any the expression
is comparable to the -norm. Further in the case , this expression defines a norm if . Hence for the weak spaces are
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s .
A major result that uses the -spaces is the Marcinkiewicz interpolation, Marcinkiewicz interpolation theorem, which has broad applications to
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
and the study of singular integrals.
Weighted spaces
As before, consider a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
. Let be a measurable function. The -weighted space is defined as , where means the measure defined by
or, in terms of the Radon–Nikodym theorem, Radon–Nikodym derivative, the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
for is explicitly
As -spaces, the weighted spaces have nothing special, since is equal to . But they are the natural framework for several results in harmonic analysis ; they appear for example in the Muckenhoupt weights, Muckenhoupt theorem: for , the classical Hilbert transform is defined on where denotes the unit circle and the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on . Muckenhoupt's theorem describes weights such that the Hilbert transform remains bounded on and the maximal operator on .
spaces on manifolds
One may also define spaces on a manifold, called the intrinsic spaces of the manifold, using Density on a manifold, densities.
Vector-valued spaces
Given a measure space and a locally-convex space , one may also define a spaces of -integrable E-valued functions in a number of ways. The most common of these being the spaces of Bochner integral, Bochner integrable and Pettis integral, Pettis-integrable functions. Using the Topological tensor product, tensor product of locally convex spaces, these may be respectively defined as
and
; where
and
respectively denote the projective and injective tensor products of locally convex spaces. When is a nuclear space, Alexander Grothendieck, Grothendieck showed that these two constructions are indistinguishable.
See also
*
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Notes
References
* .
* .
* .
* .
*
* .
* .
*
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External links
*
Proof that ''L''''p'' spaces are complete
{{DEFAULTSORT:Lp Space
Normed spaces
Banach spaces
Mathematical series
Function spaces
Measure theory