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

A sequence in a set is just an -valued map whose value at is denoted by instead of the usual parentheses notation

### Space of all sequences

Let denote the field either of real or complex numbers. The product denotes the set of all sequences of scalars in This set can become a vector space when vector addition is defined by

and the scalar multiplication is defined by

A **sequence space** is any linear subspace of

As a topological space, is naturally endowed with the product topology. Under this topology, is Fréchet, meaning that it is a complete, metrizable, locally convex topological vector space (TVS). However, this topology is rather pathological: there are no continuous norms on (and thus the product topology cannot be defined by any norm).^{[1]} Among Fréchet spaces, is minimal in having no continuous norms:

**Theorem ^{[1]}** — Let be a Fréchet space over
Then the following are equivalent:

- admits no continuous norm (that is, any continuous seminorm on has a nontrivial null space).
- contains a vector subspace TVS-isomorphic to .
- contains a complemented vector subspace TVS-isomorphic to .

But the product topology is also unavoidable: does not admit a strictly coarser Hausdorff, locally convex topology.^{[1]} For that reason, the study of sequences begins by finding a strict linear subspace of interest, and endowing it with a topology *different* from the subspace topology.

### ℓ^{p} spaces

For is the subspace of consisting of all sequences satisfying

If then the real-valued operation defined by

defines a norm on In fact, is a complete metric space with respect to this norm, and therefore is a Banach space.

If then does not carry a norm, but rather a metric defined by

If then is defined to be the space of all bounded sequences endowed with the norm

is also a Banach space.

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

The space of convergent sequences *c* is a sequence space. This consists of all such that lim_{n→∞} *x*_{n} exists. Since every convergent sequence is bounded, *c* is a linear subspace of . 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 where for the first entries (for ) and is zero everywhere else (i.e. ) is Cauchy, but does not converge to a sequence in *c*_{00}.

### Space of all finite sequences

Let

- ,

denote the **space of finite sequences over** . As a vector space, is equal to , but has a different topology.

For every natural number , let denote the usual Euclidean space endowed with the Euclidean topology and let denote the canonical inclusion

- .

The image of each inclusion is

and consequently,

This family of inclusions gives a final topology , defined to be the finest topology on such that all the inclusions are continuous (an example of a coherent topology). With this topology, becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is *not* Fréchet–Urysohn. The topology is also strictly finer than the subspace topology induced on by .

Convergence in has a natural description: if and is a sequence in then in if and only is eventually contained in a single image and under the natural topology of that image.

Often, each image is identified with the corresponding ; explicitly, the elements and are identified. This is facilitated by the fact that the subspace topology on , the quotient topology from the map , and the Euclidean topology on all coincide. With this identification, is the direct limit of the directed system where every inclusion adds trailing zeros:

- .

This shows is an LB-space.

### Other sequence spaces

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

This space, when equipped with the norm

is a Banach space isometrically isomorphic to 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}

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*

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 .

## Properties of ℓ^{1} spaces

A sequence of elements in ℓ^{1} converges in the space of complex sequences ℓ^{1} if and only if it converges weakly in this space.^{[2]}
If *K* is a subset of this space, then the following are equivalent:^{[2]}

*K*is compact;*K*is weakly compact;*K*is bounded, closed, and equismall at infinity.

Here *K* being **equismall at infinity** means that for every , there exists a natural number such that for all .

## See also

## References

- ^
^{a}^{b}^{c}Jarchow 1981, pp. 129-130. - ^
^{a}^{b}Trèves 2006, pp. 451-458.

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