In mathematics, the Lp 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 (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910).

Lp 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.

Applications

edit

Statistics

edit

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be defined in terms of   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   norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared   norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero).[1] Elastic net regularization uses a penalty term that is a combination of the   norm and the squared   norm of the parameter vector.

Hausdorff–Young inequality

edit

The Fourier transform for the real line (or, for periodic functions, see Fourier series), 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.

By contrast, if   the Fourier transform does not map into  

Hilbert spaces

edit

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. 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 p-norm in finite dimensions

edit
 
Illustrations of unit circles (see also superellipse) in   based on different  -norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding  ).

The Euclidean length of a vector   in the  -dimensional real vector space   is given by the Euclidean norm:

 

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 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, 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, physics, and computer science.

Definition

edit

For a real number   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.

The  -norm or maximum norm (or uniform norm) is the limit of the  -norms for   It turns out that this limit is equivalent to the following definition:

 

See L-infinity.

For all   the  -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 (positive homogeneity), 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   together with the  -norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. This Banach space is the  -space over  

Relations between p-norms

edit

The grid distance or rectilinear distance (sometimes called the "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  -norms in that the  -norm   of any given vector   does not grow with  :

  for any vector   and real numbers   and   (In fact this remains true for   and   .)

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.

In general, for vectors in   where  

 

This is a consequence of Hölder's inequality.

When 0 < p < 1

edit
 
Astroid, unit circle in   metric

In   for   the formula

 
defines an absolutely homogeneous function for   however, the resulting function does not define a norm, because it is not subadditive. 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. 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 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 scalar multiple   of the  -unit ball contains the convex hull of   which is equal to   The fact that for fixed   we have

 
shows that the infinite-dimensional sequence space   defined below, is no longer locally convex.[citation needed]

When p = 0

edit

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. The space of sequences has a complete metric topology provided by the F-norm

 
which is discussed by Stefan Rolewicz in Metric Linear Spaces.[2] The  -normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the   "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  [citation needed] Many authors abuse terminology by omitting the quotation marks. Defining   the zero "norm" of   is equal to

 
 
An animated gif of p-norms 0.1 through 2 with a step of 0.05.

This is not a norm because it is not homogeneous. 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, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis. Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances.

The p-norm in infinite dimensions and p spaces

edit

The sequence space p

edit

The  -norm can be extended to vectors that have an infinite number of components (sequences), which yields the space  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 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  -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,   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:

 
and the corresponding space   of all bounded sequences. It turns out that[3]
 
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. 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 instead of a sum is used to define the  -norm.

General ℓp-space

edit

In complete analogy to the preceding definition one can define the space   over a general index set   (and  ) as

 
where convergence on the right means that only countably many summands are nonzero (see also Unconditional convergence). 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 direct limit of  -sequence spaces.[4]

For   the  -norm is even induced by a canonical inner product   called the Euclidean inner product, which means that   holds for all vectors   This inner product can expressed in terms of the norm by using the polarization identity. On   it can be defined by

 
while for the space   associated with a measure space   which consists of all square-integrable functions, it is
 

Now consider the case   Define[note 1]

 
where for all  [5][note 2]
 

The index set   can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space   is just a special case of the more general  -space (defined below).

Lp spaces and Lebesgue integrals

edit

An   space may be defined as a space of measurable functions for which the  -th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let   be a measure space and  [note 3] When  , consider the set   of all measurable functions   from   to   or   whose absolute value raised to the  -th power has a finite integral, or in symbols:

 

To define the set for   recall that two functions   and   defined on   are said to be equal almost everywhere, written   a.e., if the set   is measurable and has measure zero. Similarly, a measurable function   (and its absolute value) is bounded (or dominated) almost everywhere by a real number   written   a.e., if the (necessarily) measurable set   has measure zero. The space   is the set of all measurable functions   that are bounded almost everywhere (by some real  ) and   is defined as the infimum of these bounds:

 
When   then this is the same as the essential supremum of the absolute value of  :[note 4]
 

For example, if   is a measurable function that is equal to   almost everywhere[note 5] then   for every   and thus   for all  

For every positive   the value under   of a measurable function   and its absolute value   are always the same (that is,   for all  ) and so a measurable function belongs to   if and only if its absolute value does. Because of this, many formulas involving  -norms are stated only for non-negative real-valued functions. Consider for example the identity   which holds whenever   is measurable,   is real, and   (here   when  ). The non-negativity requirement   can be removed by substituting   in for   which gives   Note in particular that when   is finite then the formula   relates the  -norm to the  -norm.

Seminormed space of  -th power integrable functions

Each set of functions   forms a vector space when addition and scalar multiplication are defined pointwise.[note 6] That the sum of two  -th power integrable functions   and   is again  -th power integrable follows from  [proof 1] although it is also a consequence of Minkowski's inequality

 
which establishes that   satisfies the triangle inequality for   (the triangle inequality does not hold for  ). That   is closed under scalar multiplication is due to   being absolutely homogeneous, which means that   for every scalar   and every function  

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm. Thus   is a seminorm and the set   of  -th power integrable functions together with the function   defines a seminormed vector space. In general, the seminorm   is not a norm because there might exist measurable functions   that satisfy   but are not identically equal to  [note 5] (  is a norm if and only if no such   exists).

Zero sets of  -seminorms

If   is measurable and equals   a.e. then   for all positive   On the other hand, if   is a measurable function for which there exists some   such that   then   almost everywhere. When   is finite then this follows from the   case and the formula   mentioned above.

Thus if   is positive and   is any measurable function, then   if and only if   almost everywhere. Since the right hand side (  a.e.) does not mention   it follows that all   have the same zero set (it does not depend on  ). So denote this common set by

 
This set is a vector subspace of   for every positive  

Quotient vector space

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

Given any   the coset   consists of all measurable functions   that are equal to   almost everywhere. The set of all cosets, typically denoted by

 
forms a vector space with origin   when vector addition and scalar multiplication are defined by   and   This particular quotient vector space will be denoted by  

Two cosets are equal   if and only if   (or equivalently,  ), which happens if and only if   almost everywhere; if this is the case then   and   are identified in the quotient space.

The  -norm on the quotient vector space

Given any   the value of the seminorm   on the coset   is constant and equal to   denote this unique value by   so that:

 
This assignment   defines a map, which will also be denoted by   on the quotient vector space
 
This map is a norm on   called the  -norm. The value   of a coset   is independent of the particular function   that was chosen to represent the coset, meaning that if   is any coset then   for every   (since   for every  ).

The Lebesgue   space

The normed vector space   is called   space or the Lebesgue space of  -th power integrable functions and it is a Banach space for every   (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem). When the underlying measure space   is understood then   is often abbreviated   or even just   Depending on the author, the subscript notation   might denote either   or  

If the seminorm   on   happens to be a norm (which happens if and only if  ) then the normed space   will be linearly isometrically isomorphic to the normed quotient space   via the canonical map   (since  ); in other words, they will be, up to a linear isometry, the same normed space and so they may both be called "  space".

The above definitions generalize to Bochner spaces.

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.

Special cases

edit

Similar to the   spaces,   is the only Hilbert space 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 and quantum mechanics. Functions in   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 (Titchmarsh 1976).

If we use complex-valued functions, the space   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   defines a bounded operator on any   space by multiplication.

For   the   spaces are a special case of   spaces, when   consists of the natural numbers and   is the counting measure 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 Lp spaces

edit

As in the discrete case, if there exists   such that   then[citation needed]

 

Hölder's inequality

Suppose   satisfy   (where  ). If   and   then   and[6]

 

This inequality, called Hölder's inequality, is in some sense optimal[6] since if   (so  ) and   is a measurable function such that

 
where the supremum is taken over the closed unit ball of   then   and
 

Minkowski inequality

Minkowski inequality, which states that   satisfies the triangle inequality, can be generalized: If the measurable function   is non-negative (where   and   are measure spaces) then for all  [7]

 

Atomic decomposition

edit

If   then every non-negative   has an atomic decomposition,[8] meaning that there exist a sequence   of non-negative real numbers and a sequence of non-negative functions   called the atoms, whose supports   are pairwise disjoint sets of measure   such that

 
and for every integer  
 
and
 
and where moreover, the sequence of functions   depends only on   (it is independent of  ).[8] These inequalities guarantee that   for all integers   while the supports of   being pairwise disjoint implies[8]
 

An atomic decomposition can be explicitly given by first defining for every integer  [8]

 
(this infimum is attained by   that is,   holds) and then letting
 
where   denotes the measure of the set   and   denotes the indicator function of the set   The sequence   is decreasing and converges to   as  [8] Consequently, if   then   and   so that   is identically equal to   (in particular, the division   by   causes no issues).

The complementary cumulative distribution function   of   that was used to define the   also appears in the definition of the weak  -norm (given below) and can be used to express the  -norm   (for  ) of   as the integral[8]

 
where the integration is with respect to the usual Lebesgue measure on  

Dual spaces

edit

The dual space (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 by the extremal case of Hölder's inequality. It is also possible to show (for example with the Radon–Nikodym theorem, see[9]) that any   can be expressed this way: i.e., that   is onto. Since   is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say simply that   is the continuous dual 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 (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 with respect to   See ba space for more details. If we assume the axiom of choice, this space is much bigger than   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   is  [10]

Embeddings

edit

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.[11] Suppose that   Then:

  1.   if and only if   does not contain sets of finite but arbitrarily large measure (any finite measure, for example).
  2.   if and only if   does not contain sets of non-zero but arbitrarily small measure (the counting measure, for example).

Neither condition holds for the real line with the Lebesgue measure while both conditions holds for the counting measure on any finite set. 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 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 of the identity   is precisely

 
the case of equality being achieved exactly when    -almost-everywhere.

Dense subspaces

edit

Throughout this section we assume that  

Let   be a measure space. An integrable simple function   on   is one of the form

 
where   are scalars,   has finite measure and   is the indicator function of the set   for   By construction of the integral, the vector space of integrable simple functions is dense in  

More can be said when   is a normal topological space and   its Borel 𝜎–algebra, i.e., the smallest 𝜎–algebra of subsets of   containing the open sets.

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
 

Closed subspaces

edit

If   is any positive real number,   is a probability measure on a measurable space   (so that  ), and   is a vector subspace, then   is a closed subspace of   if and only if   is finite-dimensional[12] (  was chosen independent of  ). In this theorem, which is due to Alexander Grothendieck,[12] it is crucial that the vector space   be a subset of   since it is possible to construct an infinite-dimensional closed vector subspace of   (that is even a subset of  ), where   is Lebesgue measure on the unit circle   and   is the probability measure that results from dividing it by its mass  [12]

Lp (0 < p < 1)

edit

Let   be a measure space. If   then   can be defined as above: it is the quotient 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 (Rudin 1991, §1.47)

 
and so the function
 
is a metric on   The resulting metric space is complete;[13] the verification is similar to the familiar case when   The balls
 
form a local base at the origin for this topology, as   ranges over the positive reals.[13] These balls satisfy   for all real   which in particular shows that   is a bounded neighborhood of the origin;[13] in other words, this space is locally bounded, just like every normed space, despite   not being a norm.

In this setting   satisfies a reverse Minkowski inequality, that is for  

 

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

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 the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: 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 (Rudin 1991, §1.47). As a particular consequence, there are no nonzero continuous linear functionals on   the continuous dual space is the zero space. In the case of the counting measure 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 Hp 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 Hp for   (Duren 1970, §7.5).

L0, the space of measurable functions

edit

The vector space of (equivalence classes of) measurable functions on   is denoted   (Kalton, Peck & Roberts 1984). 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 space   is always a topological abelian group but is only a topological vector space if   This is because scalar multiplication is continuous if and only if   If   is  -finite then the weaker topology of local convergence in measure is an F-space, i.e. a completely metrizable topological vector space. Moreover, this topology is isometric to global convergence in measure   for a suitable choice of probability measure  

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 Lévy-metric for   Under this metric the space   is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if  . To see this, consider the Lebesgue measurable function   defined by  . Then clearly  . 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