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 only exists 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, that is a linear subspace of vector space of all complex valued sequences, equipped with the topology of pointwise convergence.

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Examples

of all complex valued sequences is trivially an FK-space.

Properties

with the topology of pointwise convergence the inclusion map is a continuous function.

FK-space constructions

a countable family of seminorms is again an FK-space.

See also

• − FK-spaces with a normable topology

References