Beta-dual space

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

Definition

Given a sequence space Template:Mvar, the Template:Math-dual of Template:Mvar is defined as

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

Here, $𝕂\in \{ℝ,ℂ\}$ so that $𝕂$ denotes either the real or complex scalar field.

If Template:Mvar is an FK-space then each Template:Mvar in Template:Math defines a continuous linear form on Template:Mvar

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

Examples

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

Properties

The beta-dual of an FK-space Template:Mvar is a linear subspace of the continuous dual of Template:Mvar. If Template:Mvar is an FK-AK space then the beta dual is linear isomorphic to the continuous dual.