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 [math]\displaystyle{ X }[/math], 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 [math]\displaystyle{ X }[/math] as [math]\displaystyle{ \left(x_n\right)_{n \in \N} }[/math] with [math]\displaystyle{ x_n \in \Complex }[/math].
Then sequence [math]\displaystyle{ \left(a_n\right)_{n \in \N}^{(k)} }[/math] in [math]\displaystyle{ X }[/math] converges to some point [math]\displaystyle{ \left(x_n\right)_{n \in \N} }[/math] if it converges pointwise for each [math]\displaystyle{ n. }[/math] That is [math]\displaystyle{ \lim_{k \to \infty} \left(a_n\right)_{n \in \N}^{(k)} = \left(x_n\right)_{n \in \N} }[/math] if for all [math]\displaystyle{ n \in \N, }[/math] [math]\displaystyle{ \lim_{k \to \infty} a_n^{(k)} = x_n }[/math]
Examples
The sequence space [math]\displaystyle{ \omega }[/math] of all complex valued sequences is trivially an FK-space.
Properties
Given an FK-space [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \omega }[/math] with the topology of pointwise convergence the inclusion map [math]\displaystyle{ \iota : X \to \omega }[/math] is a continuous function.
FK-space constructions
Given a countable family of FK-spaces [math]\displaystyle{ \left(X_n, P_n\right) }[/math] with [math]\displaystyle{ P_n }[/math] a countable family of seminorms, we define [math]\displaystyle{ X := \bigcap_{n=1}^{\infty} X_n }[/math] and [math]\displaystyle{ P := \left\{p_{\vert X} : p \in P_n\right\}. }[/math] Then [math]\displaystyle{ (X,P) }[/math] is again an FK-space.
See also
- BK-space – Sequence space that is Banach − FK-spaces with a normable topology
- FK-AK space
- Sequence space – Vector space of infinite sequences
References
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