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 $$X$$, 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 $$\left(x_n\right)_{n \in \N}$$ with $$x_n \in \Complex$$.

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

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

Properties
Given an FK-space $$X$$ 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.