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Lp space
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In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding sequence spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to they were first introduced by . They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.
ider the real vector space Rn.
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In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding sequence spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to they were first introduced by . They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.
Motivation
Consider the real vector space Rn. The sum of two vectors in Rn is given by
and the scalar action is given by
The length of a vector x = (x1, x2, …, xn) is usually given by the Euclidean norm
but this is by no means the only way of defining length. If p is a real number, p = 1, define the Lp norm of x by
(so the L2 norm is the familiar Euclidean norm, while the distance in the L1 norm is known as the Manhattan distance).
One also extends this to p = 8 via
which is in fact the limit of the p norms for finite p.
The L8 norm is also known as the uniform norm.
It turns out that for all p = 1 this definition indeed satisfies 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, and
- the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).
For any p = 1, Rn together with the Lp norm (or simply p-norm) just defined becomes a Banach space.
When 0 < p < 1
In Rn for n > 1, the formula
makes sense for 0 < p < 1, though the resulting function does not define a norm, because it violates the triangle inequality. However, the function
defines a metric. The metric space (Rn, dp) is denoted by lnp.
Although the p-unit ball Bnp around the origin in this metric is "concave", the topology defined on Rn by the metric dp is the usual vector space topology of Rn, hence lnp is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of lnp is to denote by Cp(n) the smallest constant C such that the multiple C Bnp of the p-unit ball contains the convex hull of Bnp, equal to Bn1. The fact that Cp(n) = n1/p – 1 tends to infinity with n (for fixed p < 1) reflects the fact that the infinite-dimensional sequence space lp defined below, is no longer locally convex.
p=0
In discrete mathematics, the p-norm is also extended to p = 0 via
which is the number of non-zero entries of the vector x. Defining 00 = 0, the zero norm of x is equal to . This is not a norm in the usual sense, but can be used to measure sparseness, e.g. in Compressed sensing.
lp spaces
The above p-norm can be extended to vectors having an infinite number of components, yielding the space lp. This contains as special cases:
The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate.
Explicitly, for an infinite sequence of real (or complex) numbers, define the vector sum to be
while the scalar action is given by
Define the p-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, (1, 1, 1, ...), will have an infinite p-norm (length) for every finite p = 1. The space lp is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite.
One can check that as p increases, the set lp grows larger. For example, the sequence
is not in l1, but it is in lp for p > 1, as the series
diverges for p = 1 (the harmonic series), but is convergent for p > 1.
One also defines the 8-norm as
and the corresponding space l8 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 lp spaces for 1 = p = 8.
The p-norm thus defined on lp is indeed a norm, and lp together with this norm is a Banach space. The fully general Lp 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 p-norm.
Properties of lp spaces and the space c0
The space l2 is the only lp space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram identity . Substituting two distinct unit vectors in for x and y directly shows that the identity is not true unless p = 2.
If 1 < p < 8, then the (continuous) dual space of lp is lq, where q is the Hölder conjugate of p: 1/p + 1/q = 1. Symbolically, (lp)* = lq. As a consequence lp is reflexive, because (lp)** = (lq)* = lp (for a more detailed study of duality, see the section "Dual of Lp" below, that also applies here).
The space c0 is defined as the space of all sequences converging to zero, with norm identical to ||x||8. It is a closed subspace of l8, hence a Banach space. The dual of c0 is l1; the dual of l1 is l8. For the case of natural numbers index set, the lp and c0 are separable, with the sole exception of l8. The dual of l8 is the ba space.
The spaces c0 and lp (for 1 = p < 8) have a canonical Schauder basis , where ei is the sequence which is zero but for a 1 in the ith entry.
The lp spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some lp or of c0, was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space of l1, has an answer in the affirmative (easy and old).
Except for the trivial finite dimensional case, an unusual feature of lp is that it is not polynomially reflexive.
Lp spaces
Let 1 = p < 8 and (S, S, µ) be a measure space. Consider the set of all measurable functions from S to C (or R) whose absolute value raised to the p-th power has a finite Lebesgue integral, or equivalently, that
The set of such functions forms a vector space, with the following natural operations:
for every scalar ?.
That the sum of two pth power integrable functions is again pth power integrable follows from the inequality |f + g|p = 2p (|f|p + |g|p). In fact, more is true. Minkowski's inequality says the triangle inequality holds for || . ||p. Thus the set of pth power integrable functions, together with the function || . ||p, is a seminormed vector space, which is denoted by .
This can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the kernel of || · ||p. Since for any measurable function f, we have that ||f||p = 0 if and only if f = 0 almost everywhere, the kernel of || . ||p does not depend upon p,
In the quotient space, two functions f and g are identified if f = g almost everywhere. The resulting normed vector space is, by definition,
For p = 8, the space L8(S, µ) is defined as follows. We start with the set of all measurable functions from S to C (or R) which are essentially bounded, i.e. bounded up to a set of measure zero. Again two such functions are identified if they are equal almost everywhere. Denote this set by L8(S, µ). For f in L8(S, µ), its essential supremum serves as an appropriate norm:
As before, we have
if f ? L8(S, µ) n Lq(S, µ) for some q < 8.
For 1 = p = 8, Lp(S, µ) is a Banach space. The fact that Lp is complete is often referred to as Riesz-Fischer theorem. Completeness can be checked using the convergence theorems for Lebesgue integrals.
When the underlying measure space S is understood, Lp(S, µ) is often abbreviated Lp(µ), or just Lp. The above definitions generalize to Bochner spaces.
Special cases
When p = 2; like the l2 space, the space L2 is the only Hilbert space of this class. In the complex case, the inner product on L2 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 L2 are sometimes called square-integrable or square-summable, but sometimes these terms are reserved for functions which are square-integrable in some other sense (such as in the sense of a Riemann integral).
If we use complex-valued functions, the space L8 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 L8 defines a bounded operator on any Lp space by multiplication.
The lp spaces (1 = p = 8) are a special case of Lp spaces, when S is the set N of positive integers, and the measure µ is the counting measure on N. More generally, if one considers any set S with the counting measure, the resulting L p space is denoted lp(S). For example, the space lp(Z) is the space of all sequences indexed by the integers, and when defining the p-norm on such a space, one sums over all the integers. The space lp(n), where n is the set with n elements, is Rn with its p-norm as defined above. As any Hilbert space, every space L2 is linearly isometric to a suitable l2(I), where the cardinality of the set I is the cardinality of an arbitrary Hilbertian basis for this particular L2.
Properties of Lp spaces
Dual spaces
The dual space (the space of all continuous linear functionals) of Lp(µ) for 1 < p < 8 has a natural isomorphism with Lq(µ), where q is such that 1/p + 1/q = 1, which associates g ? Lq(µ) with the functional ?p(g) ? Lp(µ)* defined by
The fact that ?p(g) is well defined and continuous follows from Hölder's inequality. The mapping ?p is a linear mapping from Lq(µ) into Lp(µ)*, 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) that any G ? Lp(µ)* can be expressed this way: i.e., that ?p is onto. Since ?p is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say shortly that Lq "is" the dual of Lp.
When 1 < p < 8, the space Lp(µ) is reflexive. Let ?p be the above map and let ?q be the corresponding linear isometry from Lp(µ) onto Lq(µ)*. The map
from Lp(µ) to Lp(µ)**, obtained by composing ?q with the transpose (or adjoint) of the inverse of ?p,
coincides with the canonical embedding J of Lp(µ) into its bidual. Moreover, the map jp is onto, as composition of two onto isometries, and this proves reflexivity.
If the measure µ on S is sigma-finite, then the dual of L1(µ) is isometrically isomorphic to L8(µ) (more precisely, the map ?1 corresponding to p = 1 is an isometry from L8(µ) onto L1(µ)*). However, except in rather trivial cases, the dual of L8 is much bigger than L1. Elements of (L8)* can be identified with bounded signed finitely additive measures on S in a construction similar to the ba space.
Embeddings
Colloquially, if 1 = p < q = 8, Lp(S, µ) contains functions that are more locally singular, while elements of Lq(S, µ) can be more spread out. Consider the Lebesgue measure on the half line (0, 8). A continuous function in L1 might blow up near 0 but must decay sufficiently fast toward infinity. On the other hand, continuous functions in L8 need not decay at all but no blow-up is allowed. The precise technical result is the following:
- Let 1 = p < q = 8. Lq(S, µ) is contained in Lp(S, µ) iff S does not contain sets of arbitrarily large measure, and
- Let 1 = p < q = 8. Lp(S, µ) is contained in Lq(S, µ) iff S does not contain sets of arbitrarily small measure.
In particular, if the domain S has finite measure, the bound (a consequence of Hölder's inequality)
means the space Lq is continuously embedded in Lp.
Dense subspaces
It is assumed that 1 = p < 8 throughout this section.
Let (S, S, µ) be a measure space. An integrable simple function f on S is one of the form
where aj is scalar and Aj ? S has finite measure, for j = 1,...,n. By construction of the integral, the vector space of integrable simple functions is dense in Lp(S, S, µ).
More can be said when S is a metrizable topological space and S its Borel s–algebra, i.e., the smallest s–algebra of subsets of S containing the open sets.
Suppose that V ? S is an open set with µ(V) < 8. It can be proved that for every Borel set A ? S contained in V, and for every e > 0, there exist a closed set F and an open set U such that
It follows that there exists f continuous on S such that
If S can be covered by an increasing sequence (Vn) of open sets that have finite measure, then the space of p–integrable continuous functions is dense in Lp(S, S, µ). More precisely, one can use bounded continuous functions that vanish outside one of the open sets Vn.
This applies in particular when S = Rd and when µ is the Lebesgue measure. The space of continuous and compactly supported functions is dense in Lp(Rd). Similarly, the space of integrable step functions is dense in Lp(Rd); this space is the linear span of indicator functions of bounded intervals when d = 1, of bounded rectangles when d = 2 and more generally of products of bounded intervals.
Several properties of general functions in Lp(Rd) 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 Lp(Rd), in the following sense: for every f ? Lp(Rd),
when t ? Rd tends to 0, where is the translated function defined by
Applications
Lp spaces are widely used in mathematics and applications.
Hausdorff-Young inequality
The Fourier transform for the real line (resp. for periodic functions, cf. Fourier series) maps Lp(R) to Lq(R) (resp. Lp(T) to lq), where 1 = p = 2 and 1/p + 1/q = 1. This is a consequence of the Riesz-Thorin interpolation theorem, and is made precise with the Hausdorff-Young inequality.
By contrast, if p > 2, the Fourier transform does not map into Lq.
Hilbert spaces
Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L2 and l2 are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to l2(E), where E is a set with an appropriate cardinality.
Statistics
In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of Lp metrics, and measures of central tendency can be characterized as solutions to variational problems.
Lp for 0 < p < 1
Let (S, S, µ) be a measure space. If 0 < p < 1, then Lp(µ) can be defined as above: it is the vector space of those measurable functions f such that
As before, we may introduce the p-norm || f ||p = Np(f)1/p,
but || · ||p does not satisfy the triangle inequality in this case, and defines only a quasi-norm.
The inequality (a + b)p = ap + bp, valid for a = 0 and b = 0 implies that
and so the function
is a metric on Lp(µ). The resulting metric space is complete; the verification is similar to the familiar case when p = 1.
In this setting Lp satisfies a reverse Minkowski inequality, that is for u and v in Lp
.
This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces Lp
for 1 < p < 8 .
The space Lp for 0 < p < 1 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 p = 1. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in lp or
Lp([0, 1]), every open convex set containing the 0 function is unbounded for the p-quasi-norm; therefore, the 0 vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space S contains an infinite family of disjoint measurable sets of finite positive measure.
The only nonempty convex open set in ''L''''p''([0, 1]) is the entire space . As a particular consequence, there are no nonzero linear functionals on ''L''''p''([0, 1]): the dual space is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space ''L''''p''(''µ'') = l''p''), the bounded linear functionals on l''p'' are exactly those that are bounded on l1, namely those given by sequences in l8. Although l''p'' 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 R''n'', rather than work with ''L''''p'' for 0 < ''p'' < 1, it is common to work with the Hardy space ''H''''p'' 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 ''H''''p'' for ''p'' < 1 .
''L''0, the space of measurable functions
The vector space of (equivalence classes of) measurable functions on (''S'', ''S'', ''µ'') is denoted ''L''0(''S'', ''S'', ''µ'') by many authors. It clearly contains all the ''L''''p'', and is equipped with the topology of ''convergence in measure'' (named ''convergence in probability'' in the special case of a probability measure ''µ'', i.e. ''µ''(''S'') = 1).
The description is easier when ''µ'' is finite.
If ''µ'' is a finite measure on (''S'', ''S''), the 0 function admits for the convergence in measure the following fundamental system of neighborhoods
The topology can be defined by any metric ''d'' of the form
where ''f'' is bounded continuous concave and non-decreasing on [0, 8), with ''f''(0) = 0 and ''f''(''t'') > 0 when ''t'' > 0 (for example, ''f''(''t'') = min(''t'', 1)). Such a metric is called ''Lévy''-''metric for'' ''L''0. Under this metric the space ''L''0 is complete (it is again an F-space). The space ''L''0 is in general not locally bounded, and not locally convex.
For the infinite Lebesgue measure ''?'' on R''n'', the definition of the fundamental system of neighborhoods could be modified as follows
The resulting space ''L''0(R''n'', ''?'') coincides as topological vector space with ''L''0(R''n'', ''g''(''x'') d''?''(x)), for any positive ''?''–integrable density ''g''.
Weak ''Lp''
Let (''S'', ''S'', ''µ'') be a measure space, and ''f'' a measurable function with real or complex values on ''S''. The distribution function of ''f'' is defined for ''t'' > 0 by
If ''f'' is in ''L''''p''(''S'', ''µ'') for some ''p'' with 1 = ''p'' < 8, then by Markov's inequality,
A function ''f'' is said to be in the space weak ''Lp''(''S'', ''µ''), or ''Lp,w''(''S'', ''µ''), if there is a constant ''C'' > 0 such that, for all ''t'' > 0,
The best constant ''C'' for this inequality is the ''Lp,w''-norm of ''f'', and is denoted by
The weak ''L''''p'' coincide with the Lorentz spaces ''L''''p'',8, so this notation is also used to denote them.
The ''Lp,w''-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for ''f'' in ''L''p(''S'', ''µ''),
and in particular ''Lp''(''S'', ''µ'') ? ''Lp,w''(''S'', ''µ''). Under the convention that two functions are equal if they are equal ''µ'' almost everywhere, then the spaces ''L''p,w are complete .
For any 0 < ''r'' < ''p'' the expression
is comparable to the ''Lp,w''-norm. Further in the case ''p'' > 1, this expression defines a norm if ''r'' = 1. Hence for ''p'' > 1 the weak ''L''''p'' spaces are Banach spaces .
A major result that uses the ''Lp,w''-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.
Weighted ''Lp'' spaces
As before, consider a measure space (''S'', ''S'', ''µ''). Let be a measurable function. The ''w''-weighted ''Lp'' space is defined as ''Lp''(''S'', ''w'' d''µ''), where ''w'' d''µ'' means the measure ''?'' defined by
or, in terms of the Radon-Nikodym derivative,
The norm for ''Lp''(''S'', ''w'' d''µ'') is explicitly
As ''L''''p''-spaces, the weighted spaces have nothing special, since ''Lp''(''S'', ''w'' d''µ'') is equal to ''L''''p''(''S'', ''?''). But they are the natural framework for several results in Harmonic Analysis; they appear for example in the Muckenhoupt theorem: for 1 < ''p'' < 8, the classical Hilbert transform is defined on ''L''''p''(T, ''?'') where T denotes the unit circle and ''?'' the Lebesgue measure; the (nonlinear) Hardy-Littlewood maximal operator is bounded on ''L''''p''(R''n'', ''?''). Muckenhoupt's theorem describes weights ''w'' such that the Hilbert transform remains bounded on ''Lp''(T, ''w'' d''?'') and the maximal operator on ''Lp''(R''n'', ''w'' d''?'').
See also
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