# 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

[edit]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

[edit]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:

- 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

[edit]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*, *c*_{0} and *c*_{00}

[edit]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

[edit]

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

[edit]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}

[edit]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*

[edit]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 .

*ℓ*^{2} is isomorphic to all separable, infinite dimensional Hilbert spaces

[edit]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:

- If H is infinite dimensional, then it is isomorphic to
*ℓ*^{2} - If dim(
*H*) =*N*, then H is isomorphic to

## Properties of *ℓ*^{1} spaces

[edit]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.^{[3]}
If *K* is a subset of this space, then the following are equivalent:^{[3]}

*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

[edit]## References

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

## Bibliography

[edit]- Banach, Stefan; 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. - Jarchow, Hans (1981).
*Locally convex spaces*. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. - 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. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - 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. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.