Sequence space

The most important sequence spaces in analysis are the ⁠${\displaystyle {\ell }^p}$⁠ spaces, consisting of the ⁠${\displaystyle p}$⁠-power summable sequences, with the ⁠${\displaystyle p}$⁠-norm. These are special cases of ⁠${\displaystyle L^p}$⁠ 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 ⁠${\displaystyle c}$⁠ and ⁠${\displaystyle c_0}$⁠, 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 ${\displaystyle x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}}$ in a set ⁠${\displaystyle X}$⁠ is an ⁠${\displaystyle X}$⁠-valued map ${\displaystyle x_{\bullet }:\mathbb{N}\to X}$ whose value at ⁠${\displaystyle n\in \mathbb{N}}$⁠ is denoted by ⁠${\displaystyle x_n}$⁠ instead of the usual parentheses notation ⁠${\displaystyle x(n)}$⁠.

Space of all sequences

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

A sequence space is any linear subspace of ⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠.

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

Theorem—Let ⁠${\displaystyle X}$⁠ be a Fréchet space over ⁠${\displaystyle \mathbb{K}}$⁠. Then the following are equivalent:

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

But the product topology is also unavoidable: ⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ 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.

ℓp spaces

For ⁠${\displaystyle 0<p<\mathrm{\infty }}$⁠, ⁠${\displaystyle {\ell }^p}$⁠ is the subspace of ⁠${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$⁠ consisting of all sequences ${\displaystyle x_{\bullet }=(x_n{)}_{n\in \mathbb{N}}}$ satisfying $${\displaystyle \sum _n|x_n{|}^p<\mathrm{\infty }.}$$

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

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

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

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

c, c0 and c00

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

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

The space of eventually zero sequences, ⁠${\displaystyle c_{00}}$⁠, is the subspace of ⁠${\displaystyle c_0}$⁠ 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 ${\displaystyle (x_{nk}{)}_{k\in \mathbb{N}}}$ where ${\displaystyle x_{nk}=1/k}$ for the first ${\displaystyle n}$ entries (for ${\displaystyle k=1,\dots ,n}$) and is zero everywhere else (that is, ${\displaystyle (x_{nk}{)}_{k\in \mathbb{N}}=}$${\displaystyle (1,\frac{1}{2},\dots ,}$${\displaystyle \frac{1}{n-1},\frac{1}{n},}$${\displaystyle 0,0,\dots )}$) is a Cauchy sequence but it does not converge to a sequence in ${\displaystyle c_{00}.}$

Space of all finite sequences

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

denote the space of finite sequences over ⁠${\displaystyle \mathbb{K}}$⁠. As a vector space, ${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$ is equal to ⁠${\displaystyle c_{00}}$⁠, but ⁠${\displaystyle {\mathbb{K}}^{\mathrm{\infty }}}$⁠ has a different topology.

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

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

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

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

Other sequence spaces

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

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

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

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

The space ⁠${\displaystyle \mathrm{\Phi }}$⁠ or ${\displaystyle c_{00}}$ 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.