Beta-dual space
Encyclopedia
In functional analysis
and related areas of mathematics
, the beta-dual or
-dual is a certain linear subspace of the algebraic dual of a sequence space
.
the 
-dual of 
is defined as
If
is an FK-space
then each
in 
defines a continuous linear form on 
of the continuous dual of E. If E is an FK-AK space
then the beta dual is linear isomorphic to the continuous dual.
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...
, the beta-dual or
-dual is a certain linear subspace of the algebraic dual of 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...
.
Definition
Given a sequence space the -dual of is defined asIf
is an FK-spaceFK-space
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....
then each
in defines a continuous linear form on Examples
Properties
The beta-dual of an FK-space E 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 the continuous dual of E. If E is an FK-AK space
FK-AK space
In functional analysis and related areas of mathematics an FK-AK space or FK-space with the AK property is an FK-space which contains the space of finite sequences and has a Schauder basis.-Examples and non-examples:...
then the beta dual is linear isomorphic to the continuous dual.