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.

- Definition
- Space of all sequences
- ℓp spaces
- c, c0 and c00
- Space of all finite sequences
- Other sequence spaces
- Properties of ℓp spaces and the space c0
- ℓp spaces are increasing in p
- Properties of ℓ1 spaces
- See also
- References
- Bibliography

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.

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

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 this regard:

**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.

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.

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}.

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.

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.

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.

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 .

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 .

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, a **normed vector space** or **normed space** is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

- It is nonnegative, that is for every vector x, one has
- It is positive on nonzero vectors, that is,
- For every vector x, and every scalar one has
- The triangle inequality holds; that is, for every vectors x and y, one has

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In mathematics, the ** L^{p} spaces** are function spaces defined using a natural generalization of the

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is *not* reflexive but is nevertheless isometrically isomorphic to its bidual.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In functional analysis, a **bounded linear operator** is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces, then is bounded if and only if there exists some such that for all in

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In mathematics, a **nuclear space** is a topological vector space that can be viewed as a generalization of finite dimensional Euclidean spaces that is different from Hilbert spaces. Nuclear spaces have many of the desirable properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In functional analysis, the **dual norm** is a measure of size for a continuous linear function defined on a normed vector space.

In the branch of mathematics called functional analysis, when a topological vector space admits a direct sum decomposition ≅, the spaces and are called *complements* of each other. This happens if and only if the addition map which is defined by is a homeomorphism. Note that while this addition map is always continuous, it may fail to be a homeomorphism, which is why this definition is needed.

The theorem on the **surjection of Fréchet spaces** is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective.

The strongest locally convex topological vector space (TVS) topology on , the tensor product of two locally convex TVSs, making the canonical map continuous is called the **projective topology** or the **π-topology**. When *X ⊗ Y* is endowed with this topology then it is denoted by and called the **projective tensor product** of *X* and *Y*.

In mathematics, the **injective tensor product** of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the *completed injective tensor products*. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with*out* any need to extend definitions from real/complex-valued functions to Y-valued functions.

In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

In mathematics, an ** LB-space**, also written

In the mathematical discipline of functional analysis, it is possible to generalize the notion of derivative to arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-value function is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. All vector spaces will be assumed to be over the field where is either the real numbers or the complex numbers

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

In mathematical analysis, the **spaces of test functions and distributions** are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the *canonical LF-topoogy*, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called * the space of distributions on * and is denoted by where the "" subscript indicates that the continuous dual space of denote by is endowed with the strong dual topology.

- 1 2 3 Jarchow 1981, pp. 129-130.
- 1 2 Trèves 2006, pp. 451-458.

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