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.¹ FK-spaces with a normable topology are called BK-spaces.

There only exists 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.

A FK-space is a sequence space of ${\displaystyle X}$, 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 $${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$$ with ${\displaystyle x_{n}\in \mathbb {C} }$.

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

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

Given an FK-space of ${\displaystyle X}$ and ${\displaystyle \omega }$ with the topology of pointwise convergence the inclusion map $${\displaystyle \iota :X\to \omega }$$ is a continuous function.

Given a countable family of FK-spaces ${\displaystyle \left(X_{n},P_{n}\right)}$ with ${\displaystyle P_{n}}$ a countable family of seminorms, we define $${\displaystyle X:=\bigcap _{n=1}^{\infty }X_{n}}$$ and $${\displaystyle P:=\left\{p_{\vert X}:p\in P_{n}\right\}.}$$ Then ${\displaystyle (X,P)}$ is again an FK-space.