Lp space
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In mathematics, the  spaces are function spaces defined using a natural generalization of the -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. Applications

Statistics

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are 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  norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero. Techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small. Elastic net regularization uses a penalty term that is a combination of the  norm and the  norm of the parameter vector. Hausdorff–Young inequality

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

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 -norm in finite dimensions

The length of a vector  in the -dimensional real vector space  is usually 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 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, 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

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 -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 -norms

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 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 

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. 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. The space 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—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

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 -norm in infinite dimensions and  spaces

The sequence space 

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 is 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

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

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. For  the -norm is even induced by a canonical inner product  called the , 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

where for all 

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

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  
When  is real (that is, ), 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 , written , if the set  is measurable and has measure zero. Similarly, a measurable function  (and its absolute value) is  (or )  by a real number  written , 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 :
 

For example, if  is a measurable function that is equal to  almost everywhere 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. That the sum of two -th power integrable functions  and  is again -th power integrable follows from  
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  equal to  ( 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  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 . 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  or the  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#Completeness of Lp, 0 < p ≤ ∞|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

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 . 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

As in the discrete case, if there exists  such that  then

Hölder's inequality

Suppose  satisfy  (where ). If  and  then  and 

This inequality, called Hölder's inequality, is in some sense optimal 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 then for all

Atomic decomposition

If  then every non-negative  has an , meaning that there exist a sequence  of non-negative real numbers and a sequence of non-negative functions  called , 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 ). These inequalities guarantee that  for all integers  while the supports of  being pairwise disjoint implies 

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

(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  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

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

Dual spaces

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) 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

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 (any finite measure, for example). 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

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 –algebra, i.e., the smallest –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

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 ( was chosen independent of ). In this theorem, which is due to Alexander Grothendieck, 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

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 

and so the function

is a metric on  The resulting metric space is complete; 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. These balls satisfy  for all real  which in particular shows that  is a bounded neighborhood of the origin; 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  . 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 . 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  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 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 with real or complex values on  The 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 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 spaces . A major result that uses the -spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals. Weighted  spaces

As before, consider a measure space  Let  be a measurable function. The -weighted  space is defined as  where  means the measure  defined by

or, in terms of the Radon–Nikodym derivative,  the norm 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 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 densities. Vector-valued  spaces

Given a measure space  and a locally convex space  (here assumed to be complete), it is possible to define spaces of -integrable -valued functions on  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 that are (each in their own way) a natural generalization of the usual  topology. Another way involves topological tensor products of  with  Element of the vector space  are finite sums of simple tensors  where each simple tensor  may be identified with the function  that sends  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, denoted by  and the injective tensor product, denoted by  In general, neither of these space are complete so their completions are constructed, which are respectively denoted by  and  (this is analogous to how the space of scalar-valued simple functions on  when seminormed by any  is not complete so a completion is constructed which, after being quotiented by  is isometrically isomorphic to the Banach space ). Alexander Grothendieck showed that when  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. See also

Notes

References

 . . . . . . External links

 
 Proof that Lp spaces are complete

Banach spaces
Function spaces
Mathematical series
Measure theory
Normed spaces
Lp spaces