FK-space

From Wikipedia, the free encyclopedia

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.

Contents

[edit] Definition

A FK-space is a sequence space 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 X as

(x_n)_{n\in\mathbb{N}} with x_n \in \mathbb{C}

Then sequence (a_n)_{n\in\mathbb{N}}^{(k)} in X converges to some point (x_n)_{n\in\mathbb{N}} if it converges pointwise for each n. That is

\lim_{k \to \infty} (a_n)_{n\in\mathbb{N}}^{(k)} = (x_n)_{n\in\mathbb{N}}

if

\forall n \in \mathbb{N} : \lim_{k \to \infty} a_n^{(k)} = x_n

[edit] Examples

[edit] Properties

Given an FK-space X and ω with the topology of pointwise convergence the inclusion map

\iota:X \to \omega

is continuous.

[edit] FK-space constructions

Given a countable family of FK-spaces (Xn,Pn) with Pn a countable family of semi-norms, we define

X:=\bigcap_{n=1}^{\infty} X_n

and

P:=\{p_{\vert X} \mid p \in P_n \}.

Then (X,P) is again an FK-space.

[edit] See also