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

$$\begin{gathered} X^{\beta }:=\{x\in {\mathbb{𝕂}}^{\mathbb{\mathbb{N}}} \\ : \\ \sum _{i=1}^{\mathrm{\infty }}{x}_i{y}_i\text{ converges }\mathrm{\forall }y\in X\}. \end{gathered}$$

Here, $\mathbb{𝕂}\in \{\mathbb{ℝ},\mathbb{ℂ}\}$ so that $\mathbb{𝕂}$ 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

$f_y(x):=\sum _{i=1}^{\mathrm{\infty }}{x}_i{y}_{i}x\in X.$

Examples

• $c_0^{\beta }={\ell }^1$
• $({\ell }^1{)}^{\beta }={\ell }^{\mathrm{\infty }}$
• ${\omega }^{\beta }=\{0\}$

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.