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.

See also

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