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

[math]\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 \}. }[/math]

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

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

Examples

• [math]\displaystyle{ c_0^\beta = \ell^1 }[/math]
• [math]\displaystyle{ (\ell^1)^\beta = \ell^\infty }[/math]
• [math]\displaystyle{ \omega^\beta = \{0\} }[/math]

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.