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 ${\displaystyle X}$, 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 ${\displaystyle X}$ as

${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ with ${\displaystyle x_{n}\in \mathbb {C} }$

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

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

if

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

Examples

• The sequence space ${\displaystyle \omega }$ of all complex valued sequences is trivially an FK-space.

Properties

Given an FK-space ${\displaystyle X}$ and ${\displaystyle \omega }$ with the topology of pointwise convergence the inclusion map

${\displaystyle \iota :X\to \omega }$

is continuous.

FK-space constructions

Given a countable family of FK-spaces ${\displaystyle (X_{n},P_{n})}$ with ${\displaystyle P_{n}}$ a countable family of semi-norms, we define

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

and

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

Then ${\displaystyle (X,P)}$ is again an FK-space.