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