FK-space

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

$${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$$

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 ${\displaystyle X}$ 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.

• BK-space – Sequence space that is Banach − FK-spaces with a normable topology
• FK-AK space
• Sequence space – Vector space of infinite sequences