FK-space

Infunctional analysisand related areas ofmathematicsaFK-spaceorFréchet coordinate spaceis asequence spaceequipped with atopological structuresuch that it becomes aFréchet space.FK-spaces with anormable topologyare calledBK-spaces.

There exists only one topology to turn a sequence space into aFréchet space,namely thetopology of pointwise convergence.Thus the namecoordinate spacebecause a sequence in an FK-space converges if and only if it converges for each coordinate.

FK-spaces are examples oftopological vector spaces.They are important insummability theory.

Definition

AFK-spaceis asequence spaceof$X$,that is alinear subspaceof vector space of all complex valued sequences, equipped with the topology ofpointwise convergence.

We write the elements of$X$as $$\left(x_{n}\right)_{n\in \mathbb {N} }$$with$x_{n}\in \mathbb {C}$.

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

Examples

The sequence space$\omega$of allcomplex valued sequencesis trivially an FK-space.

Properties

Given an FK-space of$X$and$\omega$with the topology of pointwise convergence theinclusion map $$\iota:X\to \omega$$ is acontinuous function.

FK-space constructions

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

See also

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