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 p spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp 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 c0, 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.


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 set of all sequences of elements of is a vector space for componentwise addition

and componentwise scalar multiplication

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:

  1. admits no continuous norm (that is, any continuous seminorm on has a nontrivial null space).
  2. contains a vector subspace TVS-isomorphic to .
  3. 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.

p spaces

See also: Lp space and L-infinity

For is the subspace of consisting of all sequences satisfying

If then the real-valued function on 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 is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all by

The canonical norm induced by this inner product is the usual -norm, meaning that for all

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

is also a Banach space.

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

c, c0 and c00

See also: c space

A convergent sequence is any sequence such that exists. The set of all convergent sequences is a vector subspace of called the space of convergent sequences. Since every convergent sequence is bounded, is a linear subspace of Moreover, this sequence space is a closed subspace of with respect to the supremum norm, and so it is a Banach space with respect to this norm.

A sequence that converges to is called a null sequence and is said to vanish. The set of all sequences that converge to is a closed vector subspace of that when endowed with the supremum norm becomes a Banach space that is denoted by and is called the space of null sequences or the space of vanishing sequences.

The space of eventually zero sequences, is the subspace of consisting 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 (that is, ) is a Cauchy sequence but it does not converge to a sequence in

Space of all finite sequences



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 ℓp spaces and the space c0

See also: c space

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 law

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

Each p is distinct, in that p is a strict subset of s whenever p < s; furthermore, p is not linearly isomorphic to s when ps. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from s to 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 s, and is thus said to be strictly singular.

If 1 < p < ∞, then the (continuous) dual space of ℓp is isometrically isomorphic to ℓq, where q is the Hölder conjugate of p: 1/p + 1/q = 1. The specific isomorphism associates to an element x of q the functional

for y in p. Hölder's inequality implies that Lx is a bounded linear functional on p, and in fact
so that the operator norm satisfies

In fact, taking y to be the element of p with

gives Lx(y) = ||x||q, so that in fact

Conversely, given a bounded linear functional L on p, the sequence defined by xn = L(en) lies in ℓq. Thus the mapping gives an isometry

The map

obtained by composing κp with the inverse of its transpose coincides with the canonical injection of ℓq into its double dual. As a consequence ℓq is a reflexive space. By abuse of notation, it is typical to identify ℓq with the dual of ℓp: (ℓp)* = ℓq. Then reflexivity is understood by the sequence of identifications (ℓp)** = (ℓq)* = ℓp.

The space c0 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 c0 is ℓ1; the dual of ℓ1 is ℓ. For the case of natural numbers index set, the ℓp and c0 are separable, with the sole exception of ℓ. The dual of ℓ is the ba space.

The spaces c0 and ℓp (for 1 ≤ p < ∞) have a canonical unconditional Schauder basis {ei | i = 1, 2,...}, where ei is the sequence which is zero but for a 1 in the i th entry.

The space ℓ1 has the Schur property: In ℓ1, 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 ℓ1 that are weak convergent but not strong convergent.

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, 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 ℓ1, that is, there does not exist a subspace Y of ℓ1 such that . In fact, ℓ1 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 ℓp is isomorphic to any other, there are thus uncountably many ker Q's).

Except for the trivial finite-dimensional case, an unusual feature of ℓp is that it is not polynomially reflexive.

p spaces are increasing in p

For , the spaces are increasing in , with the inclusion operator being continuous: for , one has . Indeed, the inequality is homogeneous in the , so it is sufficient to prove it under the assumption that . In this case, we need only show that for . But if , then for all , and then .

2 is isomorphic to all separable, infinite dimensional Hilbert spaces

Let H be a separable Hilbert space. Every orthogonal set in H is at most countable (i.e. has finite dimension or ).[2] The following two items are related:

Properties of 1 spaces

A sequence of elements in 1 converges in the space of complex sequences 1 if and only if it converges weakly in this space.[3] If K is a subset of this space, then the following are equivalent:[3]

  1. K is compact;
  2. K is weakly compact;
  3. 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


  1. ^ a b c Jarchow 1981, pp. 129–130.
  2. ^ Debnath, Lokenath; Mikusinski, Piotr (2005). Hilbert Spaces with Applications. Elsevier. pp. 120–121. ISBN 978-0-12-2084386.
  3. ^ a b Trèves 2006, pp. 451–458.