 FK-space
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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

$(x_{n})_{n\in \mathbb {N} }$ with $x_{n}\in \mathbb {C}$ Then sequence $(a_{n})_{n\in \mathbb {N} }^{(k)}$ in $X$ converges to some point $(x_{n})_{n\in \mathbb {N} }$ if it converges pointwise for each $n$ . That is

$\lim _{k\to \infty }(a_{n})_{n\in \mathbb {N} }^{(k)}=(x_{n})_{n\in \mathbb {N} }$ if

$\forall n\in \mathbb {N} :\lim _{k\to \infty }a_{n}^{(k)}=x_{n}$ ## 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

$\iota :X\to \omega$ 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

$X:=\bigcap _{n=1}^{\infty }X_{n}$ and

$P:=\{p_{\vert X}\mid p\in P_{n}\}$ .

Then $(X,P)$ is again an FK-space.