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In
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
a FK-space or Fréchet coordinate space is a
sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural nu ...
equipped with a
topological structure In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
such that it becomes a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
. FK-spaces with a normable topology are called BK-spaces. There exists only one topology to turn a sequence space into a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
, namely the
topology of pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
. 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. They are important in summability theory.


Definition

A FK-space is a
sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural nu ...
X, that is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of vector space of all complex valued sequences, equipped with the topology of
pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set an ...
. We write the elements of X as \left(x_n\right)_ with x_n \in \Complex. Then sequence \left(a_n\right)_^ in X converges to some point \left(x_n\right)_ if it converges pointwise for each n. That is \lim_ \left(a_n\right)_^ = \left(x_n\right)_ if for all n \in \N, \lim_ a_n^ = 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 In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota ...
\iota : X \to \omega is a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
.


FK-space constructions

Given a countable family of FK-spaces \left(X_n, P_n\right) with P_n a countable family of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s, we define X := \bigcap_^ X_n and P := \left\. Then (X,P) is again an FK-space.


See also

* − FK-spaces with a normable topology * *


References

{{DEFAULTSORT:Fk-Space F-spaces Fréchet spaces Topological vector spaces