Sequence Space

Get Sequence Space essential facts below. View Videos or join the Sequence Space discussion. Add Sequence Space to your PopFlock.com topic list for future reference or share this resource on social media.
## Definition

### l^{p} spaces

*c*, *c*_{0} and *c*_{00}

### Other sequence spaces

## Properties of l^{p} spaces and the space *c*_{0}

### l^{p} spaces are increasing in *p*

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Sequence Space

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field **K** of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in **K**, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

The most important sequence spaces in analysis are the l^{p} spaces, consisting of the *p*-power summable sequences, with the *p*-norm. These are special cases of L^{p} spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted *c* and *c*_{0}, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.

Let **K** denote the field either of real or complex numbers. Denote by **K**^{N} the set of all sequences of scalars

This can be turned into a vector space by defining vector addition as

and the scalar multiplication as

A **sequence space** is any linear subspace of **K**^{N}.

For 0 < *p* < ?, l^{p} is the subspace of **K**^{N} consisting of all sequences *x* = (**x**_{n}) satisfying

If *p* >= 1, then the real-valued operation defined by

defines a norm on l^{p}. In fact, l^{p} is a complete metric space with respect to this norm, and therefore is a Banach space.

If 0 < *p* < 1, then l^{p} does not carry a norm, but rather a metric defined by

If *p* = ?, then l^{?} is defined to be the space of all bounded sequences. With respect to the norm

l^{?} is also a Banach space.

The space of convergent sequences *c* is a sequence space. This consists of all *x* ? **K**^{N} such that lim_{n->?}*x*_{n} exists. Since every convergent sequence is bounded, *c* is a linear subspace of l^{?}. It is, moreover, a closed subspace with respect to the infinity norm, and so a Banach space in its own right.

The subspace of null sequences *c*_{0} consists of all sequences whose limit is zero. This is a closed subspace of *c*, and so again a Banach space.

The subspace of eventually zero sequences *c*_{00} consists of all sequences which have only finitely many nonzero elements. This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm. For example, the sequence (*x*_{n}) where *x*_{n} has 1/*k* for the first *n* entries and is zero everywhere else (i.e. *x*_{n} = (1, 1/2, ..., 1/*n*, 0, ...)) is Cauchy w.r.t. infinity norm but not convergent (to a sequence in *c*_{00}).

The space of bounded series, denote by bs, is the space of sequences *x* for which

This space, when equipped with the norm

is a Banach space isometrically isomorphic to l^{?}, via the linear mapping

The subspace *cs* consisting of all convergent series is a subspace that goes over to the space *c* under this isomorphism.

The space ? or is defined to be the space of all infinite sequences with only a finite number of non-zero terms (sequences with finite support). This set is dense in many sequence spaces.

The space l^{2} is the only l^{p} space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law

Substituting two distinct unit vectors for *x* and *y* directly shows that the identity is not true unless *p* = 2.

Each l^{p} is distinct, in that l^{p} is a strict subset of l^{s} whenever *p* < *s*; furthermore, l^{p} is not linearly isomorphic to l^{s} when *p* ? *s*. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from l^{s} to l^{p} is compact when *p* < *s*. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of l^{s}, and is thus said to be strictly singular.

If 1 < *p* < ?, then the (continuous) dual space of l^{p} is isometrically isomorphic to l^{q}, where *q* is the Hölder conjugate of *p*: 1/*p* + 1/*q* = 1. The specific isomorphism associates to an element *x* of l^{q} the functional

for *y* in l^{p}. Hölder's inequality implies that *L*_{x} is a bounded linear functional on l^{p}, and in fact

so that the operator norm satisfies

In fact, taking *y* to be the element of l^{p} with

gives *L*_{x}(*y*) = ||*x*||_{q}, so that in fact

Conversely, given a bounded linear functional *L* on l^{p}, the sequence defined by *x*_{n} = *L*(*e*_{n}) lies in l^{q}. Thus the mapping gives an isometry

The map

obtained by composing ?_{p} with the inverse of its transpose coincides with the canonical injection of l^{q} into its double dual. As a consequence l^{q} is a reflexive space. By abuse of notation, it is typical to identify l^{q} with the dual of l^{p}: (l^{p})^{*} = l^{q}. Then reflexivity is understood by the sequence of identifications (l^{p})^{**} = (l^{q})^{*} = l^{p}.

The space *c*_{0} is defined as the space of all sequences converging to zero, with norm identical to ||*x*||_{?}. It is a closed subspace of l^{?}, hence a Banach space. The dual of *c*_{0} is l^{1}; the dual of l^{1} is l^{?}. For the case of natural numbers index set, the l^{p} and *c*_{0} are separable, with the sole exception of l^{?}. The dual of l^{?} is the ba space.

The spaces *c*_{0} and l^{p} (for 1 p < ?) have a canonical unconditional Schauder basis {*e*_{i} | *i* = 1, 2,...}, where *e*_{i} is the sequence which is zero but for a 1 in the *i*^{th} entry.

The space l^{1} has the Schur property: In l^{1}, any sequence that is weakly convergent is also strongly convergent (Schur 1921). However, since the weak topology on infinite-dimensional spaces is strictly weaker than the strong topology, there are nets in l^{1} that are weak convergent but not strong convergent.

The l^{p} spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some l^{p} or of *c*_{0}, 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 l^{1}, was answered in the affirmative by Banach & Mazur (1933). That is, for every separable Banach space *X*, there exists a quotient map , so that *X* is isomorphic to . In general, ker *Q* is not complemented in l^{1}, that is, there does not exist a subspace *Y* of l^{1} such that . In fact, l^{1} has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ; since there are uncountably many such *X* 's, and since no l^{p} is isomorphic to any other, there are thus uncountably many ker *Q* 's).

Except for the trivial finite-dimensional case, an unusual feature of l^{p} is that it is not polynomially reflexive.

For , the spaces are increasing in , with the inclusion operator being continuous: for , one has .

This follows from defining for , and noting that for all , which can be shown to imply .

- Banach, S.; Mazur, S. (1933), "Zur Theorie der linearen Dimension",
*Studia Mathematica*,**4**: 100-112. - Dunford, Nelson; Schwartz, Jacob T. (1958),
*Linear operators, volume I*, Wiley-Interscience. - Pitt, H.R. (1936), "A note on bilinear forms",
*J. London Math. Soc.*,**11**(3): 174-180, doi:10.1112/jlms/s1-11.3.174. - Schur, J. (1921), "Über lineare Transformationen in der Theorie der unendlichen Reihen",
*Journal für die reine und angewandte Mathematik*,**151**: 79-111, doi:10.1515/crll.1921.151.79.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists