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.
A FK-space is a sequence space of
, 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
Then sequence
converges to some point
if it converges pointwise for each
lim
lim
The sequence space
ω
of all complex valued sequences is trivially an FK-space.
Given an FK-space of
ω
with the topology of pointwise convergence the inclusion map
ι :
is a continuous function.
Given a countable family of FK-spaces
a countable family of seminorms, we define