# 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.

## Definition

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}.

### ℓ^{p} spaces

^{p}spaces

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.

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

*c*,

*c*

_{0}and

*c*

_{00}

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}).

### Other sequence spaces

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 Φ 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.

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

^{p}spaces and the space

*c*

_{0}

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 κ_{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.

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

^{p}spaces are increasing in

*p*

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 .

## References

- 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.