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.