FK-space

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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[edit]

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.

We write the elements of as

with

Then sequence in converges to some point if it converges pointwise for each . That is

if

Examples[edit]

Properties[edit]

Given an FK-space and with the topology of pointwise convergence the inclusion map

is continuous.

FK-space constructions[edit]

Given a countable family of FK-spaces with a countable family of semi-norms, we define

and

.

Then is again an FK-space.

See also[edit]