In functional analysis 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 ...

, 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.

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 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 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, li ...

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