FK-space

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 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 $${\displaystyle \underset{k\to \mathrm{\infty }}{lim}{\left(a_n\right)}_{n\in \mathbb{N}}^{(k)}={\left(x_n\right)}_{n\in \mathbb{N}}}$$ if for all ${\displaystyle n\in \mathbb{N},}$ $${\displaystyle \underset{k\to \mathrm{\infty }}{lim}{a}_n^{(k)}=x_n}$$