FK-space

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 $x_{n}\in \mathbb {C}$ .

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 $\lim _{k\to \infty }\left(a_{n}\right)_{n\in \mathbb {N} }^{(k)}=\left(x_{n}\right)_{n\in \mathbb {N} }$ if for all $n\in \mathbb {N} ,$ $\lim _{k\to \infty }a_{n}^{(k)}=x_{n}$

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 $\left(X_{n},P_{n}\right)$ with $P_{n}$ a countable family of seminorms, we define $X:=\bigcap _{n=1}^{\infty }X_{n}$ and $P:=\left\{p_{\vert X}:p\in P_{n}\right\}.$ Then $(X,P)$ is again an FK-space.

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