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Encyclopedia > Lp space

In mathematics, the L^p and $ell^p$ spaces are spaces of p-power integrable functions, and corresponding sequence spaces. They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...

L^p spaces have applications in statistics, finance, and the engineering field of finite element analysis. The introduction to this article is too long. ...

1 Motivation
2 spaces
3 Properties of spaces
4 Lp spaces
5 Special cases
6 Relation to spaces
7 Properties of Lp spaces
- 7.1 Dual spaces
- 7.2 Embeddings
8 Weighted Lp spaces
9 See also
10 Reference
11 External links

Motivation

Consider the real vector space Rⁿ. The sum of vectors in Rⁿ is given by In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...

$(x_1, x_2, dots, x_n) + (y_1, y_2, dots, y_n) = (x_1+y_1, x_2+y_2, dots, x_n+y_n),$

and the scalar action is given by

$lambda(x_1, x_2, dots, x_n)=(lambda x_1, lambda x_2, dots, lambda x_n).$

The length of a vector $x=(x_1, x_2, dots, x_n)$ is usually given by

$|x|=left(x_1^2+x_2^2+dots+x_n^2right)^{1/2}$

but this is by no means the only way of defining length. If p is a real number, p≥1, define In mathematics, the real numbers may be described informally in several different ways. ...

$|x|_p=left(|x_1|^p+|x_2|^p+dots+|x_n|^pright)^{1/p}$

for any vector $x=(x_1, x_2, dots, x_n)$ . It turns out that 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 changes (modulus-)linearly when we multiply it by a scalar, and the length of the sum of two vectors is no larger than the sum of lengths of the vectors. For any p≥1, Rⁿ together with the p-norm just defined becomes a Banach space. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

$ell^p$ spaces

The above p-norm can be extended to vectors having an infinite number of components, yielding the space $ell^p$ . For $x=(x_1, x_2, dots, x_n, x_{n+1},dots)$ an infinite sequence of real (or complex) numbers, define the vector sum to be In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

$(x_1, x_2, dots, x_n, x_{n+1},dots)+(y_1, y_2, dots, y_n, y_{n+1},dots)=(x_1+y_1, x_2+y_2, dots, x_n+y_n, x_{n+1}+y_{n+1},dots),$

while the scalar action is given by

$lambda(x_1, x_2, dots, x_n, x_{n+1},dots) = (lambda x_1, lambda x_2, dots, lambda x_n, lambda x_{n+1},dots).$

Define the p-norm

$|x|_p=left(|x_1|^p+|x_2|^p+dots+|x_n|^p+|x_{n+1}|^p+dotsright)^{1/p}.$

Here, a complication arises, that being that the series on the right is not always convergent, so for example, the sequence made up of only ones, $(1, 1, 1, dots),$ will have an infinite p-norm (length), no matter what p is. The space $ell^p$ is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite. In mathematics, a series is often represented as the sum of a sequence of terms. ...

One can check that as p increases, the set $ell^p$ grows larger. For example, the sequence

$left(1, frac{1}{2}, dots, frac{1}{n}, frac{1}{n+1},dotsright)$

is not in $ell^1$ , but it is in $ell^p$ for p>1, as the series

$1^p+frac{1}{2^p} + dots + frac{1}{n^p} + frac{1}{(n+1)^p}dots$

diverges for p=1 (the harmonic series), but is convergent for p>1. See harmonic series (music) for the (related) musical concept. ...

One also defines the ∞-norm as

$|x|_infty=sup(|x_1|, |x_2|, dots, |x_n|,|x_{n+1}|, dots)$

and the corresponding space $ell^infty$ of all bounded sequences. It turns out that

$|x|_infty=lim_{ptoinfty}|x|_p$

if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider $ell^p$ spaces for 1≤p≤∞.

The p-norm thus defined on $ell^p$ is indeed a norm, and $ell^p$ together with this norm is a Banach space. The fully general L^p space, is obtained, as seen below, when one considers vectors not only with several components or with a countably infinite many components, but rather, vectors with arbitrarily many components, in other words, functions. Instead of using a sum to define the p-norm, one will use an integral. Partial plot of a function f. ... In calculus, the integral of a function is an extension of the concept of a sum. ...

Properties of $ell^p$ spaces

The space $ell^2$ is the only $ell^p$ space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram identity $|x+y|_p^2 + |x-y|_p^2= 2|x|_p^2 + 2|y|_p^2$ . Direct substitution with unit vectors results in a counter example. In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...

The $ell^p$ , 1 < p < ∞ spaces are reflexive: $(ell^p)^*=ell^q$ , where (1/p) + (1/q) = 1. This page concerns the reflexivity of a Banach space. ...

The dual of c₀ is $ell^1$ ; the dual of $ell^1$ is $ell^infty$ . For the case of natural numbers index set, the $ell^p$ and c₀ are separable, with the sole exception of $ell^{,infty}$ . Here, c₀ is defined as the space of all sequences converging zero, with norm identical to ||x||_∞. In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...

The $ell^p$ spaces can be embedded into many Banach spaces. The question of whether all Banach spaces have such an embedding was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Boris Tsirelson is an Israeli mathematician and Professor of Mathematics in the Tel Aviv University in Israel. ... In mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an l p space nor a c0 space can be embedded. ... 1974 (MCMLXXIV) was a common year starting on Tuesday. ...

Except for the trivial finite case, an unusual feature of $ell^p$ is that it is not polynomially reflexive. In mathematics, a polynomially reflexive space is a Banach space X, on which all polynomials are reflexive. ...

L^p spaces

Let p be a positive real number and let (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 In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... The integral can be interpreted as the area under a curve. ...

$|f|_p := sqrt[p!]{int |f|^p;mathrm{d}mu}<infty.$

The vector sum of two functions is given by

(f + g)(x) = f (x) + g (x),

and it follows from the inequality $|f + g|^p le 2^p |f|^p + |g|^p$ that the sum of two p^th power integrable functions is again p^th power integrable. The scalar action on a function is given by

(λ f)(x) = λ f (x).

This vector space together with the function $|cdot|_p$ is a seminormed complete vector space denoted by $mathcal{L}^p(S, mu)$ . To make it into a Banach space one considers the Kolmogorov quotient of this space, a standard procedure for spaces which are not T₀; one divides out the kernel of the norm. Thus we define $L^p(S, mu) := mathcal{L}^p(S, mu) / mathrm{ker}(|cdot|_p)$ . This means we are identifying two functions if they are equal almost everywhere. The space L^∞(S), while related, is defined differently. We start with the set of all measurable functions from S to C (or R) which are, up to a set of measure zero, bounded. By identifying two such functions if they are equal almost everywhere, we get the set L^∞(S). For f in L^∞(S), we set In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... The word kernel has several meanings in mathematics, some related to each other and some not. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

$|f|_infty := inf { Cge 0 : |f(x)| le C mbox{ for almost every } x}.$

As before, we have

$|f|_infty=lim_{ptoinfty}|f|_p$

if f ∈ L^∞(S) ∩ L^q(S) for some q < ∞.

This definition is generalized in the context of Bochner spaces. In mathematics, Bochner spaces are a generalization of the concept of Lp spaces to slightly more general domains and ranges than the initial definition. ...

Special cases

The most important case is when p = 2; like the $ell^2$ space, the space L² is the only Hilbert space of this class, having major applications to Fourier series and quantum mechanics, as well as other fields. In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Fig. ...

If we use complex-valued functions, the space L^∞ 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, since any element of L^∞ defines an operator on the Hilbert space L² by pointwise multiplication. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... C*-algebras are an important area of research in functional analysis. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...

Relation to $ell^p$ spaces

The $ell^p$ spaces (1≤p≤∞) are a special case of L^p spaces, when the set S is the positive integers, and the measure used in the integration in the definition is a counting measure. The integers are commonly denoted by the above symbol. ... In mathematics, the counting measure is an intuitive way to put a measure on any set: the size of a subset is taken to be the number of the subsets elements if this is finite, and âˆž if the subset is infinite. ...

More generally, if one considers a countable set S with the counting measure, the obtained L^p space is denoted $ell^p(S)$ . For example, the space $ell^p(mathbb Z)$ 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 $ell^p(n)$ , where n is the set with n elements, is Rⁿ with its p-norm as defined above.

Properties of L^p spaces

If 1 ≤ p ≤ ∞, then the Minkowski inequality, proved using Hölder's inequality, establishes the triangle inequality in L^p(S). Using the convergence theorems for the Lebesgue integral, one can then show that L^p(S) is complete and hence is a Banach space. (Here it is crucial that the Lebesgue integral is employed, and not the Riemann integral.) In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. ... In mathematical analysis, HÃ¶lders inequality, named after Otto HÃ¶lder, is a fundamental inequality relating Lp spaces: let S be a measure space, let 1 â‰¤ p, q â‰¤ âˆž with 1/p + 1/q = 1, let f be in Lp(S) and g be in Lq(S). ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...

Dual spaces

The dual space (the space of all continuous linear functionals) of $L p$ for $1 < p < infty$ has a natural isomorphism with $L q$ , where q is such that 1/p + 1/q = 1, which associates $gin L^q$ with the functional G $in (L^p)^*$ defined by In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ...

$G(f) = int bar{f} g ;mbox{d}mu$

(where $bar{f}$ means the complex conjugate). It is possible to show that any G $in (L^p)^*$ can be expressed this way. Since the relationship 1/p + 1/q = 1 is symmetric, L^p is reflexive for these values of p: the natural monomorphism from L^p to (L^p)^** is onto, that is, it is an isomorphism of Banach spaces. This page concerns the reflexivity of a Banach space. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

If the measure on S is sigma-finite, then the dual of L¹(S) is isomorphic to L^∞(S). However, except in rather trivial cases, the dual of L^∞ is much bigger than L¹. Elements of (L^∞)^* can be identified with bounded signed finitely additive measures on S in a construction similar to the ba space. In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the ba space of a sigma-algebra is the Banach space consisting of all bounded and finitely additive measures on . ...

If 0 < p < 1, then L^p can be defined as above, but || · ||_p does not satisfy the triangle inequality in this case, and hence it defines only a quasi-norm. However, we can still define a metric by setting d(f, g) = (||f − g||_p)^p. The resulting metric space is complete, and L^p for 0 < p < 1 is the prototypical example of an F-space that is not locally convex. In linear algebra, functional analysis and related areas of mathematics a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → R so that Scalar multiplication in V is continuous with respect to d and the standard metric on R or C. Addition in V is... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ...

Embeddings

Suppose that the domain S has finite measure, and that 1 ≤ p ≤ q ≤ ∞. Then the bound

$|f|_p le mu(S)^{(1/p)-(1/q)} |f|_q$

follows from Hölder's inequality. Hence, the space L^p is embedded in L^q.

It follows that, without any restriction on S, the embedding

$L^1_{mathrm{loc}}(S) subset L^p(S)$

holds, where the space on the left consists of the locally integrable functions on S. In mathematics, a locally integrable function is a function which is integrable on any compact set. ...

Weighted L^p spaces

As before, consider a measure space $(S, mathcal{F}, mu)$ . Let $w : S to [0, + infty)$ be an absolutely integrable function, i.e. $w in L^{1} (S, mu)$ . The $w$ -weighted L^p space is defined as $L^{p} (S, w , mathrm{d} mu)$ , where $w , mathrm{d} mu$ means the measure $ν$ defined by In mathematics, a measure is a function that assigns a number, e. ... In mathematics, an integrable function is a function whose integral exists. ...

$nu (A) := int_{A} w(x) , mathrm{d} mu (x),$

or, in terms of the Radon-Nikodym derivative, In mathematics, the Radon-Nikodym theorem is a result in functional analysis that states that if a measure Q is absolutely continuous with respect to another sigma-finite measure P then there is a measurable function f, taking values in [0,∞], on the underlying space such that for any...

$w = frac{mathrm{d} nu}{mathrm{d} mu}.$

The norm for $L^{p} (S, w , mathrm{d} mu)$ is explicitly In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

$| u |_{L^{p} (S, w , mathrm{d} mu)} := left( int_{S} w(x) | u(x) |^{p} , mathrm{d} mu (x) right)^{1/p}.$

Reference

Adams, Robert A. (1975). Sobolev Spaces. New York: Academic Press. ISBN 0-12-044150-0.

External links

Proof that L^p spaces are complete on PlanetMath

Categories: Topological vector spaces | Mathematical series PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

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spaces

Properties of spaces

Lp spaces