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 formula_1, 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 formula_1 as
Then sequence formula_5 in formula_1 converges to some point formula_3 if it converges pointwise for each formula_8. That is
if
Properties.
Given an FK-space formula_1 and formula_11 with the topology of pointwise convergence the inclusion map
is continuous.
FK-space constructions.
Given a countable family of FK-spaces formula_15 with formula_16 a countable family of semi-norms, we define
and
Then formula_19 is again an FK-space.