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 o ...

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

algebraic dual In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...

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 nu ...

Definition

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

X^:= \left \.

If is an

FK-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-spa ...

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 linear ...

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 In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...

of . If is an FK-AK space then the beta dual is linear isomorphic to the continuous dual. {{mathanalysis-stub Functional analysis