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

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

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

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

Examples

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

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.