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

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

• 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 continuous.

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.

• BK-space, FK-spaces with a normable topology