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 $X$, 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:$(x\_n)\_\{ninmathbb\{N$ with $x\_n\; in\; mathbb\{C\}$

Then sequence $(a\_n)\_\{ninmathbb\{N^\{(k)\}$ in $X$ converges to some point $(x\_n)\_\{ninmathbb\{N$ if it converges pointwise for each $n$. That is:$lim\_\{k\; o\; infty\}\; (a\_n)\_\{ninmathbb\{N^\{(k)\}\; =\; (x\_n)\_\{ninmathbb\{N$if:$forall\; n\; in\; mathbb\{N\}\; :\; lim\_\{k\; o\; infty\}\; a\_n^\{(k)\}\; =\; x\_n$

Examples

* The sequence space $omega$ of all complex valued sequences is trivially an FK-space.

Properties

Given an FK-space $X$ and $omega$ with the topology of pointwise convergence the inclusion map:$iota:X\; o\; omega$is continuous.

FK-space constructions

Given a countable family of FK-spaces $(X\_n,P\_n)$ with $P\_n$ a countable family of semi-norms, we define:and:$P:=\{p\_\{vert\; X\}\; mid\; p\; in\; P\_n\; \}$.Then $(X,P)$ is again an FK-space.

See also

*BK-space, FK-spaces with a normable topology