FK-space

Template:Short description Template:Multiple issues In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces with a normable topology are called BK-spaces.

There exists only one topology to turn a sequence space into a Fréchet space, namely the topology of pointwise convergence. Thus the name coordinate space because a sequence in an FK-space converges if and only if it converges for each coordinate.

FK-spaces are examples of topological vector spaces. They are important in summability theory.

Definition

A FK-space is a sequence space of $X$, that is a linear subspace of vector space of all complex valued sequences, equipped with the topology of pointwise convergence.

We write the elements of $X$ as $${\left(x_n\right)}_{n\in \mathbb{N}}$$ with $x_n\in ℂ$.

Then sequence ${\left(a_n\right)}_{n\in \mathbb{N}}^{(k)}$ in $X$ converges to some point ${\left(x_n\right)}_{n\in \mathbb{N}}$ if it converges pointwise for each $n.$ That is $$\underset{k\to \mathrm{\infty }}{\lim }{\left(a_n\right)}_{n\in \mathbb{N}}^{(k)}={\left(x_n\right)}_{n\in \mathbb{N}}$$ if for all $n\in \mathbb{N},$ $$\underset{k\to \mathrm{\infty }}{\lim }{a}_n^{(k)}=x_n$$

Examples

The sequence space $\omega$ of all complex valued sequences is trivially an FK-space.

Properties

Given an FK-space of $X$ and $\omega$ with the topology of pointwise convergence the inclusion map $$\iota :X\to \omega$$ is a continuous function.

FK-space constructions

Given a countable family of FK-spaces $(X_n,P_n)$ with $P_n$ a countable family of seminorms, we define $$X:=\bigcap _{n=1}^{\mathrm{\infty }}{X}_n$$ and $$P:=\{p_{|X}:p\in P_n\}.$$ Then $(X,P)$ is again an FK-space.

See also

• Template:Annotated link − FK-spaces with a normable topology
• Template:Annotated link
• Template:Annotated link