Beta-dual space

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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.

[edit] Definition

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

$X^{\beta}:=\{x \in \omega : \sum_{i=1}^{\infty} x_i y_i < \infty \quad \forall y \in X\}.$

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

$f_y(x) := \sum_{i=1}^{\infty} x_i y_i \qquad x \in X.$

[edit] Examples

$c_0^\beta = l^1$
$(l^1)^\beta = l^\infty$
$ω β = Φ$

[edit] 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.