Definition

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

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

$f\_y(x)\; :=\; sum\_\{i=1\}^\{infty\}\; x\_i\; y\_i\; qquad\; x\; in\; X.$

Examples

• $c\_0^beta\; =\; l^1$
• $(l^1)^beta\; =\; l^infty$
• $omega^beta\; =\; Phi$

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.