The Swallow's Tail

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

Definition

Given a sequence space $X$ the $\beta$-dual of $X$ is defined as

$X^{\beta }:=\{x\in X\mid \sum _{i=1}^{\mathrm{\infty }}{x}_i{y}_i<\mathrm{\infty }\mathrm{\forall }y\in X\}.$

If $X$ is an FK-space then each $y$ in $X^{\beta }$ defines a continuous linear form on $X$

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

Examples

• $c_0^{\beta }=l^1$
• $(l^1{)}^{\beta }=l^{\mathrm{\infty }}$
• ${\omega }^{\beta }=\mathrm{\varnothing }$

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.