Beta-dual space

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

Definition

Given a sequence space $X$ the $\beta$ -dual of $X$ is defined as

$X^{\beta}:=\{x \in \omega�: \sum_{i=1}^{\infty} x_i y_i < \infty \quad \forall y \in X\}.$

If $X$ is an FK-space then each $y$ in $X^{\beta}$ defines a continuous linear form on $X$

$f_y(x)�:= \sum_{i=1}^{\infty} x_i y_i \qquad x \in X.$

Examples

$c_0^\beta = l^1$
$(l^1)^\beta = l^\infty$
$\omega^\beta = \emptyset$

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.

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