FK-space
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
|
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 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 as with .
Then sequence in converges to some point if it converges pointwise for each That is if for all
Examples
[edit]The sequence space of all complex valued sequences is trivially an FK-space.
Properties
[edit]Given an FK-space of and with the topology of pointwise convergence the inclusion map is a continuous function.
FK-space constructions
[edit]Given a countable family of FK-spaces with a countable family of seminorms, we define and Then is again an FK-space.
See also
[edit]- BK-space – Sequence space that is Banach − FK-spaces with a normable topology
- FK-AK space
- Sequence space – Vector space of infinite sequences