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 with. Then sequence in X converges to some point if it converges pointwise for each n. That is if for all n \in \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 is a continuous function.

FK-space constructions

Given a countable family of FK-spaces with P_n a countable family of seminorms, we define and Then (X,P) is again an FK-space.