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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 $X$ , 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

\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 all complex valued sequences is trivially an FK-space.

Properties

Given an FK-space $X$ and $\omega$ with the topology of pointwise convergence the inclusion map

\iota :X\to \omega

is a continuous function.

FK-space constructions

Given a countable family of FK-spaces $\left(X_{n},P_{n}\right)$ with $P_{n}$ a countable family of seminorms, 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.

References

Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
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