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 defined ...

and related areas of mathematics, the beta-dual or -dual is a certain linear subspace of the

algebraic dual In 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 m ...

of 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 ...

Definition

Given a sequence space the -dual of is defined as :

X^:= \left \.

If is an FK-space then each in defines a

continuous linear form In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded line ...

on :

f_y(x) := \sum_^ x_i y_i \qquad x \in X.

Examples

c_0^\beta = \ell^1

(\ell^1)^\beta = \ell^\infty

\omega^\beta = \

Properties

The beta-dual of an FK-space is a linear subspace of the continuous dual of . If is an

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 * c_0 the space of converg ...

then the beta dual is linear isomorphic to the continuous dual. {{mathanalysis-stub Functional analysis