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 Template:Math spaces, consisting of the Template:Math-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 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.

Definition

A sequence $x_{\bullet }={\left(x_n\right)}_{n\in \mathbb{N}}$ in a set $X$ is just 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 sequencesTemplate:AnchorTemplate:Anchor

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).Template:Sfn Among Fréchet spaces, ${𝕂}^{\mathbb{N}}$ is minimal in having no continuous norms:

Template:Math theoremBut the product topology is also unavoidable: ${𝕂}^{\mathbb{N}}$ 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 $0<p<\mathrm{\infty },$ ${\ell }^p$ is the subspace of ${𝕂}^{\mathbb{N}}$ consisting of all sequences $x_{\bullet }={\left(x_n\right)}_{n\in \mathbb{N}}$ 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 Template:Visible anchor, 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 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 ${𝕂}^{\mathbb{N}}$ called the [[c space|Template:Em]]. 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 Template:Em and is said to Template:Em. 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 Template:Visible anchor and is called the Template:Em or the Template:Em.

The Template:Em, Template:Visible anchor 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 ${\left(x_{nk}\right)}_{k\in \mathbb{N}}$ where $x_{nk}=1/k$ for the first $n$ entries (for $k=1,\dots ,n$) and is zero everywhere else (that is, ${\left(x_{nk}\right)}_{k\in \mathbb{N}}=(1,1/2,\dots ,1/(n-1),1/n,0,0,\dots )$) 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 Template:Nowrap 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 Template:Em 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

$(x_n{)}_{n\in \mathbb{N}}\mapsto {\left(\sum _{i=0}^n{x}_i\right)}_{n\in \mathbb{N}}.$

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

The space Φ 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

Template:See also The space ℓ² 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

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

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

Each Template:Math is distinct, in that Template:Math is a strict subset of Template:Math whenever p < s; furthermore, Template:Math is not linearly isomorphic to Template:Math when Template:Math. In fact, by Pitt's theorem Template:Harv, every bounded linear operator from Template:Math to Template:Math is compact when Template:Math. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of Template:Math, 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 Template:Math the functional $$L_x(y)=\sum _n{x}_n{y}_n$$ for y in Template:Math. Hölder's inequality implies that L_x is a bounded linear functional on Template:Math, and in fact $$|L_x(y)|\le \Vert x{\Vert }_q\Vert y{\Vert }_p$$ so that the operator norm satisfies

$\Vert L_x{\Vert }_{({\ell }^p{)}^{*}}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{y\in {\ell }^p,y=0}{\sup }\frac{|L_x(y)|}{\Vert y{\Vert }_p}\le \Vert x{\Vert }_q.$

In fact, taking y to be the element of Template:Math 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) = ||x||_q, so that in fact

$\Vert L_x{\Vert }_{({\ell }^p{)}^{*}}=\Vert x{\Vert }_q.$

Conversely, given a bounded linear functional L on Template:Math, the sequence defined by Template:Math lies in ℓ^q. Thus the mapping $x\mapsto L_x$ gives an isometry $${\kappa }_q:{\ell }^q\to ({\ell }^p{)}^{*}.$$

The map

${\ell }^q\stackrel{{\kappa }_q}{\to }({\ell }^p{)}^{*}\stackrel{({\kappa }_q^{*}{)}^{-1}}{\to }({\ell }^q{)}^{**}$

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 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 ℓ^p and c₀ are separable, with the sole exception of ℓ^∞. The dual of ℓ^∞ is the ba space.

The spaces c₀ 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 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 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 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 Template:Harvtxt. 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 ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that ${\ell }^1=Y\oplus \ker Q$. In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take $X={\ell }^p$; since there are uncountably many such XTemplate:'s, and since no ℓ^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 ℓ^p 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 ℓ²
• If Template:Math, then H is isomorphic to ${ℂ}^N$

Properties of ℓ¹ spaces

A sequence of elements in ℓ¹ converges in the space of complex sequences ℓ¹ if and only if it converges weakly in this space.Template:Sfn If K is a subset of this space, then the following are equivalent:Template:Sfn

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

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• Template:Trèves François Topological vector spaces, distributions and kernels