Beta-dual space

In functional analysis and related areas of mathematics, the beta-dual or β-dual is a certain linear subspace of the algebraic dual of a sequence space.

Definition

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

${\displaystyle X^{\beta }:=\left\{x\in \mathbb {K} ^{\mathbb {N} }\$ :\ \sum _{i=1}^{\infty }x_{i}y_{i}{\text{ converges }}\quad \forall y\in X\right\}.}

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

If X is an FK-space then each y in X^β defines a continuous linear form on X

${\displaystyle f_{y}(x):=\sum _{i=1}^{\infty }x_{i}y_{i}\qquad x\in X.}$

Examples

• ${\displaystyle c_{0}^{\beta }=\ell ^{1}}$
• ${\displaystyle (\ell ^{1})^{\beta }=\ell ^{\infty }}$
• ${\displaystyle \omega ^{\beta }=\{0\}}$

Properties

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