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 Template:Tmath 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 Template:Tmath, 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 Template:Tmath spaces, consisting of the Template:Tmath-power summable sequences, with the Template:Tmath-norm. These are special cases of [[Lp space|Template:Tmath 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 Template:Tmath and Template:Tmath, 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 $x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}$ in a set Template:Tmath is an Template:Tmath-valued map $x_{\bullet }:\mathbb{N}\to X$ whose value at Template:Tmath is denoted by Template:Tmath instead of the usual parentheses notation Template:Tmath.

Space of all sequencesTemplate:AnchorTemplate:Anchor

Let Template:Tmath denote the field either of real or complex numbers. The set Template:Tmath of all sequences of elements of Template:Tmath is a vector space for componentwise addition $${\left(x_n\right)}_{n\in \mathbb{N}}+{\left(y_n\right)}_{n\in \mathbb{N}}={(x_n+y_n)}_{n\in \mathbb{N}},$$ and componentwise scalar multiplication $$\alpha {\left(x_n\right)}_{n\in \mathbb{N}}={\left(\alpha x_n\right)}_{n\in \mathbb{N}}.$$

A sequence space is any linear subspace of Template:Tmath.

As a topological space, Template:Tmath is naturally endowed with the product topology. Under this topology, Template:Tmath 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 Template:Tmath (and thus the product topology cannot be defined by any norm).Template:Sfn Among Fréchet spaces, Template:Tmath is minimal in having no continuous norms:

Template:Math theoremBut the product topology is also unavoidable: Template:Tmath does not admit a strictly coarser Hausdorff, locally convex topology.Template:Sfn 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.

Template:Math spaces

For Template:Tmath, Template:Tmath is the subspace of Template:Tmath consisting of all sequences $x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}$ satisfying $$\sum _n|x_n{|}^p<\mathrm{\infty }.$$

If Template:Tmath, then the real-valued function $\Vert \cdot {\Vert }_p$ on Template:Tmath defined by $$\Vert x{\Vert }_p=(\sum _n|x_n{|}^p{)}^{1/p}\text{ for all }x\in {\ell }^p$$ defines a norm on Template:Tmath. In fact, Template:Tmath is a complete metric space with respect to this norm, and therefore is a Banach space.

If Template:Tmath then Template:Tmath is also a Hilbert space when endowed with its canonical inner product, called the Template:Visible anchor, defined for all Template:Tmath by $$⟨x_{\bullet },y_{\bullet }⟩=\sum _n\overline{x_n}{y}_n.$$ The canonical norm induced by this inner product is the usual Template:Tmath-norm, meaning that $\Vert 𝐱{\Vert }_2=\sqrt{⟨𝐱,𝐱⟩}$ for all Template:Tmath.

If Template:Tmath, then Template:Tmath is defined to be the space of all bounded sequences endowed with the norm $$\Vert x{\Vert }_{\mathrm{\infty }}=\underset{n}{sup}|x_n|,$$ Template:Tmath is also a Banach space.

If Template:Tmath, then Template:Tmath does not carry a norm, but rather a metric defined by $$d(x,y)=\sum _n{|x_n-y_n|}^p.$$

Template:Math, Template:Math and Template:Math

A Template:Em is any sequence $x_{\bullet }\in {𝕂}^{\mathbb{N}}$ such that $\underset{n\to \mathrm{\infty }}{lim}{x}_n$ exists. The set Template:Visible anchor of all convergent sequences is a vector subspace of Template:Tmath called the [[c space|Template:Em]]. Since every convergent sequence is bounded, Template:Tmath is a linear subspace of Template:Tmath. Moreover, this sequence space is a closed subspace of Template:Tmath with respect to the supremum norm, and so it is a Banach space with respect to this norm.

A sequence that converges to Template:Tmath is called a Template:Em and is said to Template:Em. The set of all sequences that converge to Template:Tmath is a closed vector subspace of Template:Tmath that when endowed with the supremum norm becomes a Banach space that is denoted by Template:Visible anchor and is called the Template:Em or the Template:Em.

The Template:Em, Template:Visible anchor, is the subspace of Template:Tmath 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 $(x_{nk}{)}_{k\in \mathbb{N}}$ where $x_{nk}=1/k$ for the first $n$ entries (for $k=1,\dots ,n$) and is zero everywhere else (that is, $(x_{nk}{)}_{k\in \mathbb{N}}=$$(1,\frac{1}{2},\dots ,$$\frac{1}{n-1},\frac{1}{n},$$0,0,\dots )$) is a Cauchy sequence but it does not converge to a sequence in $c_{00}.$

Space of all finite sequences

Let $${𝕂}^{\mathrm{\infty }}=\{(x_1,x_2,\dots )\in {𝕂}^{\mathbb{N}}:\text{all but finitely many }{x}_i\text{ equal }0\}$$

denote the space of finite sequences over Template:Tmath. As a vector space, ${𝕂}^{\mathrm{\infty }}$ is equal to Template:Tmath, but Template:Tmath has a different topology.

For every natural number Template:Tmath, let Template:Tmath denote the usual Euclidean space endowed with the Euclidean topology and let ${\mathrm{In}}_{{𝕂}^n}:{𝕂}^n\to {𝕂}^{\mathrm{\infty }}$ denote the canonical inclusion $${\mathrm{In}}_{{𝕂}^n}(x_1,\dots ,x_n)=(x_1,\dots ,x_n,0,0,\dots ).$$ The image of each inclusion is $$\mathrm{Im}\left({\mathrm{In}}_{{𝕂}^n}\right)=\{(x_1,\dots ,x_n,0,0,\dots ):x_1,\dots ,x_n\in 𝕂\}={𝕂}^n\times \{(0,0,\dots )\}$$ and consequently, $${𝕂}^{\mathrm{\infty }}=\bigcup _{n\in \mathbb{N}}\mathrm{Im}\left({\mathrm{In}}_{{𝕂}^n}\right).$$

This family of inclusions gives Template:Tmath a final topology Template:Tmath, defined to be the finest topology on Template:Tmath such that all the inclusions are continuous (an example of a coherent topology). With this topology, Template:Tmath becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is Template:Em Fréchet–Urysohn. The topology Template:Tmath is also strictly finer than the subspace topology induced on Template:Tmath by Template:Tmath.

Convergence in Template:Tmath has a natural description: if $v\in {𝕂}^{\mathrm{\infty }}$ and Template:Tmath is a sequence in Template:Tmath then Template:Tmath in Template:Tmath if and only Template:Tmath is eventually contained in a single image $\mathrm{Im}\left({\mathrm{In}}_{{𝕂}^n}\right)$ and Template:Tmath under the natural topology of that image.

Often, each image $\mathrm{Im}\left({\mathrm{In}}_{{𝕂}^n}\right)$ is identified with the corresponding Template:Tmath; explicitly, the elements $(x_1,\dots ,x_n)\in {𝕂}^n$ and $(x_1,\dots ,x_n,0,0,0,\dots )$ are identified. This is facilitated by the fact that the subspace topology on Template:Nobr the quotient topology from the map Template:Nobr and the Euclidean topology on Template:Tmath all coincide. With this identification, $(({𝕂}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }}),{\left({\mathrm{In}}_{{𝕂}^n}\right)}_{n\in \mathbb{N}})$ is the direct limit of the directed system $({\left({𝕂}^n\right)}_{n\in \mathbb{N}},{\left({\mathrm{In}}_{{𝕂}^m\to {𝕂}^n}\right)}_{m\le n\in \mathbb{N}},\mathbb{N}),$ where every inclusion adds trailing zeros: $${\mathrm{In}}_{{𝕂}^m\to {𝕂}^n}(x_1,\dots ,x_m)=(x_1,\dots ,x_m,0,\dots ,0).$$ This shows $({𝕂}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }})$ is an LB-space.

Other sequence spaces

The space of bounded series, denote by bs, is the space of sequences Template:Tmath for which $$\underset{n}{sup}\left|\sum _{i=0}^n{x}_i\right|<\mathrm{\infty }.$$

This space, when equipped with the norm $$\Vert x{\Vert }_{bs}=\underset{n}{sup}\left|\sum _{i=0}^n{x}_i\right|,$$

is a Banach space isometrically isomorphic to ${\ell }^{\mathrm{\infty }},$ via the linear mapping $$(x_n{)}_{n\in \mathbb{N}}\mapsto (\sum _{i=0}^n{x}_i{)}_{n\in \mathbb{N}}.$$

The subspace $cs$ consisting of all convergent series is a subspace that goes over to the space Template:Tmath under this isomorphism.

The space Template:Tmath or $c_{00}$ 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 Template:Math spaces and the space Template:Math

Template:See also The space Template:Tmath is the only Template:Tmath space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law

$$\Vert x+y{\Vert }_p^2+\Vert x-y{\Vert }_p^2=2\Vert x{\Vert }_p^2+2\Vert y{\Vert }_p^2.$$

Substituting two distinct unit vectors for Template:Tmath and Template:Tmath directly shows that the identity is not true unless Template:Tmath.

Each Template:Tmath is distinct, in that Template:Tmath is a strict subset of Template:Tmath whenever Template:Tmath; furthermore, Template:Tmath is not linearly isomorphic to Template:Tmath when Template:Tmath. In fact, by Pitt's theorem Template:Harv, every bounded linear operator from Template:Tmath to Template:Tmath is compact when Template:Tmath. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of Template:Tmath, and is thus said to be strictly singular.

If Template:Tmath, then the (continuous) dual space of Template:Tmath is isometrically isomorphic to Template:Tmath, where Template:Tmath is the Hölder conjugate of Template:Tmath: Template:Tmath. The specific isomorphism associates to an element Template:Tmath of Template:Tmath the functional $$L_x(y)=\sum _n{x}_n{y}_n$$ for Template:Tmath in Template:Tmath. Hölder's inequality implies that Template:Tmath is a bounded linear functional on Template:Tmath, and in fact $$|L_x(y)|\le \Vert x{\Vert }_q\Vert y{\Vert }_p$$ so that the operator norm satisfies $$\Vert L_x{\Vert }_{({\ell }^p{)}^{\ast }}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{y\in {\ell }^p,y≠0}{sup}\frac{|L_x(y)|}{\Vert y{\Vert }_p}\le \Vert x{\Vert }_q.$$ In fact, taking Template:Tmath to be the element of Template:Tmath with $$y_n=\{\begin{array}{cc}0& \text{if}{x}_n=0\\ x_n^{-1}|x_n{|}^q& \text{if}{x}_n\ne 0\end{array}$$ gives Template:Nobr so that in fact $$\Vert L_x{\Vert }_{({\ell }^p{)}^{\ast }}=\Vert x{\Vert }_q.$$ Conversely, given a bounded linear functional Template:Tmath on Template:Tmath, the sequence defined by Template:Tmath lies in Template:Tmath. Thus the mapping Template:Tmath gives an isometry $${\kappa }_q:{\ell }^q\to ({\ell }^p{)}^{\ast }.$$

The map $${\ell }^q\stackrel{{\kappa }_q}{\to }({\ell }^p{)}^{\ast }\stackrel{({\kappa }_q^{\ast }{)}^{-1}}{\to }({\ell }^q{)}^{\ast \ast }$$ obtained by composing Template:Tmath with the inverse of its transpose coincides with the canonical injection of Template:Tmath into its double dual. As a consequence Template:Tmath is a reflexive space. By abuse of notation, it is typical to identify Template:Tmath with the dual of Template:Tmath: Template:Tmath. Then reflexivity is understood by the sequence of identifications Template:Tmath.

The space Template:Tmath is defined as the space of all sequences converging to zero, with norm identical to Template:Nobr It is a closed subspace of Template:Tmath, hence a Banach space. The dual of Template:Tmath is Template:Tmath; the dual of Template:Tmath is Template:Tmath. For the case of natural numbers index set, the Template:Tmath and Template:Tmath are separable, with the sole exception of Template:Tmath. The dual of Template:Tmath is the ba space.

The spaces Template:Tmath and Template:Tmath (for Template:Tmath) have a canonical unconditional Schauder basis Template:Tmath, where Template:Tmath is the sequence which is zero but for a Template:Tmath in the Template:Tmathth entry.

The space ℓ¹ has the Schur property: In ℓ¹, any sequence that is weakly convergent is also strongly convergent Template:Harv. 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 Template:Tmath spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some Template:Tmath or of Template:Tmath, 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 Template:Tmath, was answered in the affirmative by Template:Harvtxt. That is, for every separable Banach space Template:Tmath, there exists a quotient map Template:Tmath, so that Template:Tmath is isomorphic to Template:Tmath. In general, Template:Tmath is not complemented in Template:Tmath, that is, there does not exist a subspace Template:Tmath of Template:Tmath such that Template:Tmath. In fact, Template:Tmath has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take Template:Tmath; since there are uncountably many such Template:TmathTemplate:'s, and since no Template:Tmath is isomorphic to any other, there are thus uncountably many ker QTemplate:'s).

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

Template:Math spaces are increasing in Template:Mvar

For Template:Tmath, the spaces Template:Tmath are increasing in Template:Tmath, with the inclusion operator being continuous: for Template:Tmath, one has Template:Nobr Indeed, the inequality is homogeneous in the Template:Tmath, so it is sufficient to prove it under the assumption that Template:Nobr In this case, we need only show that $\sum |x_i{|}^q\le 1$ for Template:Tmath. But if Template:Nobr then $|x_i|\le 1$ for all Template:Tmath, and then $\sum |x_i{|}^q\le$Template:Nobr

Template:Math is isomorphic to all separable, infinite dimensional Hilbert spaces

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

• If Template:Tmath is infinite dimensional, then it is isomorphic to Template:Tmath,
• If Template:Tmath, then Template:Tmath is isomorphic to Template:Tmath.

Properties of Template:Math spaces

A sequence of elements in Template:Tmath converges in the space of complex sequences Template:Tmath if and only if it converges weakly in this space.Template:Sfn If Template:Tmath is a subset of this space, then the following are equivalent:Template:Sfn

1. Template:Tmath is compact;
2. Template:Tmath is weakly compact;
3. Template:Tmath is bounded, closed, and equismall at infinity.

Here Template:Tmath being equismall at infinity means that for every Template:Tmath, there exists a natural number $n_{\epsilon }\ge 0$ such that ${\sum }_{n=n_{ϵ}}^{\mathrm{\infty }}|s_n|<\epsilon$ for all Template:Tmath.

See also

• L^p space
• Tsirelson space
• beta-dual space
• Orlicz sequence space
• Hilbert space

Bibliography

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• Template:Jarchow Locally Convex Spaces
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• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
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• Template:Trèves François Topological vector spaces, distributions and kernels