Beta-dual space explained

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.

Definition

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

X ^{\beta} : = \left \{ x \in K ^{N :} \sum ^{inf ty} _{i =1} x _{i} y _{iconverges} \forall y \in X \right \}.

Here,

K \in\{ R , C \} so that K denotes either the real or complex scalar field.

If is an FK-space then each in defines a continuous linear form on

f _{y(x)} : = \sum ^{inf ty} _{i =1} x _{i} y _{i} x \in X.

Examples

c ^{\beta} _{0} = \ell ^{1} (\ell ^{1)} ^{\beta} = \ell ^{infty} \omega ^{\beta} = \{ 0 \}

Properties

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