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 ⁠${\ell }^p$⁠ 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 $x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}$ in a set ⁠$X$⁠ is an ⁠$X$⁠-valued map $x_{\bullet }:\mathbb{N}\to X$ whose value at ⁠$n\in \mathbb{N}$⁠ is denoted by ⁠$x_n$⁠ instead of the usual parentheses notation ⁠$x(n)$⁠.

Space of all sequences

Let ⁠$𝕂$⁠ denote the field either of real or complex numbers. The set ⁠${𝕂}^{\mathbb{N}}$⁠ of all sequences of elements of ⁠$𝕂$⁠ 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 ⁠${𝕂}^{\mathbb{N}}$⁠.

As a topological space, ⁠${𝕂}^{\mathbb{N}}$⁠ is naturally endowed with the product topology. Under this topology, ⁠${𝕂}^{\mathbb{N}}$⁠ 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 ⁠${𝕂}^{\mathbb{N}}$⁠ (and thus the product topology cannot be defined by any norm). Among Fréchet spaces, ⁠${𝕂}^{\mathbb{N}}$⁠ is minimal in having no continuous norms:

Theorem—Let ⁠$X$⁠ be a Fréchet space over ⁠$𝕂$⁠. Then the following are equivalent:

1. ⁠$X$⁠ admits no continuous norm (that is, any continuous seminorm on ⁠$X$⁠ has a nontrivial null space).
2. ⁠$X$⁠ contains a vector subspace TVS-isomorphic to ⁠${𝕂}^{\mathbb{N}}$⁠.
3. ⁠$X$⁠ contains a complemented vector subspace TVS-isomorphic to ⁠${𝕂}^{\mathbb{N}}$⁠.

But the product topology is also unavoidable: ⁠${𝕂}^{\mathbb{N}}$⁠ does not admit a strictly coarser Hausdorff, locally convex topology. 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 ⁠$0<p<\mathrm{\infty }$⁠, ⁠${\ell }^p$⁠ is the subspace of ⁠${𝕂}^{\mathbb{N}}$⁠ consisting of all sequences $x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}$ satisfying $$\sum _n|x_n{|}^p<\mathrm{\infty }.$$

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

If ⁠$p=2$⁠ then ⁠${\ell }^2$⁠ is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all ⁠$x_{\bullet },y_{\bullet }\in {\ell }^p$⁠ by $$⟨x_{\bullet },y_{\bullet }⟩=\sum _n\overline{x_n}{y}_n.$$ The canonical norm induced by this inner product is the usual ⁠${\ell }^2$⁠-norm, meaning that $\Vert 𝐱{\Vert }_2=\sqrt{⟨𝐱,𝐱⟩}$ for all ⁠$𝐱\in {\ell }^p$⁠.

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

If ⁠$0<p<1$⁠, then ⁠${\ell }^p$⁠ does not carry a norm, but rather a metric defined by $$d(x,y)=\sum _n{|x_n-y_n|}^p.$$

c, c₀ and c₀₀

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

A sequence that converges to ⁠$0$⁠ is called a null sequence and is said to vanish. The set of all sequences that converge to ⁠$0$⁠ is a closed vector subspace of ⁠$c$⁠ that when endowed with the supremum norm becomes a Banach space that is denoted by ⁠$c_0$⁠ and is called the space of null sequences or the space of vanishing sequences.

The space of eventually zero sequences, ⁠$c_{00}$⁠, is the subspace of ⁠$c_0$⁠ 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 ⁠$𝕂$⁠. As a vector space, ${𝕂}^{\mathrm{\infty }}$ is equal to ⁠$c_{00}$⁠, but ⁠${𝕂}^{\mathrm{\infty }}$⁠ has a different topology.

For every natural number ⁠$n\in \mathbb{N}$⁠, let ⁠${𝕂}^n$⁠ 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 ⁠${𝕂}^{\mathrm{\infty }}$⁠ a final topology ⁠${\tau }^{\mathrm{\infty }}$⁠, defined to be the finest topology on ⁠${𝕂}^{\mathrm{\infty }}$⁠ such that all the inclusions are continuous (an example of a coherent topology). With this topology, ⁠${𝕂}^{\mathrm{\infty }}$⁠ becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is not Fréchet–Urysohn. The topology ⁠${\tau }^{\mathrm{\infty }}$⁠ is also strictly finer than the subspace topology induced on ⁠${𝕂}^{\mathrm{\infty }}$⁠ by ⁠${𝕂}^{\mathbb{N}}$⁠.

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

Often, each image $\mathrm{Im}\left({\mathrm{In}}_{{𝕂}^n}\right)$ is identified with the corresponding ⁠${𝕂}^n$⁠; 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 $\mathrm{Im}\left({\mathrm{In}}_{{𝕂}^n}\right)$, the quotient topology from the map ${\mathrm{In}}_{{𝕂}^n}$, and the Euclidean topology on ⁠${𝕂}^n$⁠ 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 ⁠$x$⁠ 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 ⁠$c$⁠ under this isomorphism.

The space ⁠$\Phi$⁠ 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 ℓ^p spaces and the space c₀

The space ⁠${\ell }^2$⁠ is the only ⁠${\ell }^p$⁠ 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 ⁠$x$⁠ and ⁠$y$⁠ directly shows that the identity is not true unless ⁠$p=2$⁠.

Each ⁠${\ell }^p$⁠ is distinct, in that ⁠${\ell }^p$⁠ is a strict subset of ⁠${\ell }^s$⁠ whenever ⁠$p<s$⁠; furthermore, ⁠${\ell }^p$⁠ is not linearly isomorphic to ⁠${\ell }^s$⁠ when ⁠$p\ne s$⁠. In fact, by Pitt's theorem , every bounded linear operator from ⁠${\ell }^s$⁠ to ⁠${\ell }^p$⁠ 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 ⁠${\ell }^s$⁠, and is thus said to be strictly singular.

If ⁠$1<p<\mathrm{\infty }$⁠, then the (continuous) dual space of ⁠${\ell }^p$⁠ is isometrically isomorphic to ⁠${\ell }^q$⁠, where ⁠$q$⁠ is the Hölder conjugate of ⁠$p$⁠: ⁠$1/p+1/q=1$⁠. The specific isomorphism associates to an element ⁠$x$⁠ of ⁠${\ell }^q$⁠ the functional $$L_x(y)=\sum _n{x}_n{y}_n$$ for ⁠$y$⁠ in ⁠${\ell }^p$⁠. Hölder's inequality implies that ⁠$L_x$⁠ is a bounded linear functional on ⁠${\ell }^p$⁠, 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 ⁠$y$⁠ to be the element of ⁠${\ell }^p$⁠ 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 $L_x(y)=\Vert x{\Vert }_q$, so that in fact $$\Vert L_x{\Vert }_{({\ell }^p{)}^{\ast }}=\Vert x{\Vert }_q.$$ Conversely, given a bounded linear functional ⁠$L$⁠ on ⁠${\ell }^p$⁠, the sequence defined by ⁠$x_n=L(e_n)$⁠ lies in ⁠${\ell }^q$⁠. Thus the mapping ⁠$x\mapsto L_x$⁠ 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 ⁠${\kappa }_p$⁠ with the inverse of its transpose coincides with the canonical injection of ⁠${\ell }^q$⁠ into its double dual. As a consequence ⁠${\ell }^q$⁠ is a reflexive space. By abuse of notation, it is typical to identify ⁠${\ell }^q$⁠ with the dual of ⁠${\ell }^p$⁠: ⁠$({\ell }^p{)}^{\ast }={\ell }^q$⁠. Then reflexivity is understood by the sequence of identifications ⁠$({\ell }^p{)}^{\ast \ast }=({\ell }^q{)}^{\ast }={\ell }^p$⁠.

The space ⁠$c_0$⁠ is defined as the space of all sequences converging to zero, with norm identical to $\Vert x{\Vert }_{\mathrm{\infty }}$. It is a closed subspace of ⁠${\ell }^{\mathrm{\infty }}$⁠, hence a Banach space. The dual of ⁠$c_0$⁠ is ⁠${\ell }^1$⁠; the dual of ⁠${\ell }^1$⁠ is ⁠${\ell }^{\mathrm{\infty }}$⁠. For the case of natural numbers index set, the ⁠${\ell }^p$⁠ and ⁠$c_0$⁠ are separable, with the sole exception of ⁠${\ell }^{\mathrm{\infty }}$⁠. The dual of ⁠${\ell }^{\mathrm{\infty }}$⁠ is the ba space.

The spaces ⁠$c_0$⁠ and ⁠${\ell }^p$⁠ (for ⁠$1\le p<\mathrm{\infty }$⁠) have a canonical unconditional Schauder basis ⁠$\{e_i:i=1,2,\dots \}$⁠, where ⁠$e_i$⁠ 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 . 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 ⁠${\ell }^p$⁠ spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ⁠${\ell }^p$⁠ 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 ⁠${\ell }^1$⁠, was answered in the affirmative by . That is, for every separable Banach space ⁠$X$⁠, there exists a quotient map ⁠$Q:{\ell }^1\to X$⁠, so that ⁠$X$⁠ is isomorphic to ⁠${\ell }^1/\ker Q$⁠. In general, ⁠$\ker Q$⁠ is not complemented in ⁠${\ell }^1$⁠, that is, there does not exist a subspace ⁠$Y$⁠ of ⁠${\ell }^1$⁠ such that ⁠${\ell }^1=Y\oplus \ker Q$⁠. In fact, ⁠${\ell }^1$⁠ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ⁠$X={\ell }^p$⁠; since there are uncountably many such ⁠$X$⁠Template:'s, and since no ⁠${\ell }^p$⁠ is isomorphic to any other, there are thus uncountably many ker QTemplate:'s).

Except for the trivial finite-dimensional case, an unusual feature of ⁠${\ell }^q$⁠ is that it is not polynomially reflexive.

ℓ^p spaces are increasing in p

For ⁠$p\in [1,\mathrm{\infty }]$⁠, the spaces ⁠${\ell }^p$⁠ are increasing in ⁠$p$⁠, with the inclusion operator being continuous: for ⁠$1\le p<q\le \mathrm{\infty }$⁠, one has $\Vert x{\Vert }_q\le \Vert x{\Vert }_p$. Indeed, the inequality is homogeneous in the ⁠$x_i$⁠, so it is sufficient to prove it under the assumption that $\Vert x{\Vert }_p=1$. In this case, we need only show that $\sum |x_i{|}^q\le 1$ for ⁠$q>p$⁠. But if $\Vert x{\Vert }_p=1$, then $|x_i|\le 1$ for all ⁠$i$⁠, and then $\sum |x_i{|}^q\le$$\sum |x_i{|}^p=1$.

ℓ² 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 ⁠${\mathrm{\aleph }}_0$⁠). The following two items are related:

• If ⁠$H$⁠ is infinite dimensional, then it is isomorphic to ⁠${\ell }^2$⁠,
• If ⁠$\dim (H)=N$⁠, then ⁠$H$⁠ is isomorphic to ⁠${ℂ}^N$⁠.

Properties of ℓ¹ spaces

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

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 ⁠$\epsilon >0$⁠, there exists a natural number $n_{\epsilon }\ge 0$ such that ${\sum }_{n=n_{ϵ}}^{\mathrm{\infty }}|s_n|<\epsilon$ for all ⁠$s={\left(s_n\right)}_{n=1}^{\mathrm{\infty }}\in K$⁠.

See also

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

Bibliography

• Banach, Stefan; Mazur, S. (1933), "Zur Theorie der linearen Dimension", Studia Mathematica, 4: 100–112, doi:10.4064/sm-4-1-100-112.
• Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience.
• Template:Jarchow Locally Convex Spaces
• 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.
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• 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.
• Template:Trèves François Topological vector spaces, distributions and kernels