FK-space
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.
, 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
as
with 
Then sequence
in 
converges to some point 
if it converges pointwise for each 
. That is
if
and 
with the topology of pointwise convergence the inclusion map
is continuous.
with 
a countable family of semi-norms, we define
and
.
Then
is again an FK-space.
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a FK-space or Fréchet coordinate space is a sequence space
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 numbers to the field K of real or complex numbers...
equipped with a topological structure such that it becomes a Fréchet space
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...
. FK-spaces with a normable topology are called BK-spaces.
There exists only one topology to turn a sequence space into a Fréchet space
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...
, 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 spaceSequence 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 numbers to the field K of real or complex numbers...
, that is a linear subspaceLinear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of vector space of all complex valued sequences, equipped with the topology of pointwise convergence
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
.
We write the elements of
as with Then sequence
in converges to some point if it converges pointwise for each . That isif
Examples
- The sequence space
of all complex valued sequences is trivially an FK-space.
Properties
Given an FK-space and with the topology of pointwise convergence the inclusion mapInclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
is continuous.
FK-space constructions
Given a countable family of FK-spaces with a countable family of semi-norms, we defineand
.Then
is again an FK-space.