Lp space

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
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External links
Proof that Lp spaces are complete
Banach spaces
Function spaces
Mathematical series
Measure theory
Normed spaces
Lp spaces