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 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 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
$\left(x_n\right)_$ with $x_n \in \Complex$.

Then sequence $\left(a_n\right)_^$ in $X$ converges to some point $\left(x_n\right)_$ if it converges pointwise for each $n.$ That is
$\lim_ \left(a_n\right)_^ = \left(x_n\right)_$
if for all $n \in \N,$
$\lim_ a_n^ = x_n$

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
$\iota : X \to \omega$
is a continuous function.

FK-space constructions

Given a countable family of FK-spaces $\left(X_n, P_n\right)$ with $P_n$ a countable family of seminorms, we define
$X := \bigcap_^ X_n$
and
$P := \left\.$
Then $(X,P)$ is again an FK-space.

See also

* − FK-spaces with a normable topology
*
*

References

{{DEFAULTSORT:Fk-Space

F-spaces
Fréchet spaces
Topological vector spaces