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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]] <math>X</math>, 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 <math>X</math> as
:<math>(x_n)_{n\in\mathbb{N}}</math> with <math>x_n \in \mathbb{C}</math>
 
Then sequence <math>(a_n)_{n\in\mathbb{N}}^{(k)}</math> in <math>X</math> converges to some point <math>(x_n)_{n\in\mathbb{N}}</math> if it converges pointwise for each <math>n</math>. That is
:<math>\lim_{k \to \infty} (a_n)_{n\in\mathbb{N}}^{(k)} = (x_n)_{n\in\mathbb{N}}</math>
if
:<math>\forall n \in \mathbb{N} : \lim_{k \to \infty} a_n^{(k)} = x_n</math>
 
==Examples==
* The sequence space <math>\omega</math> of all [[space of complex valued sequences|complex valued sequences]] is trivially an FK-space.
 
==Properties==
Given an FK-space <math>X</math> and <math>\omega</math> with the topology of pointwise convergence the [[inclusion map]]
:<math>\iota:X \to \omega</math>
is [[continuous (topology)|continuous]].
 
==FK-space constructions==
Given a countable family of FK-spaces <math>(X_n,P_n)</math> with <math>P_n</math> a countable family of [[semi-norm]]s, we define
:<math>X:=\bigcap_{n=1}^{\infty} X_n</math>
and
:<math>P:=\{p_{\vert X} \mid p \in P_n \}</math>.
Then <math>(X,P)</math> is again an FK-space.
 
==See also==
*[[BK-space]], FK-spaces with a [[normable topology]]
 
{{DEFAULTSORT:Fk-Space}}
[[Category:Topological vector spaces]]

Revision as of 18:18, 4 March 2014



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