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

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

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

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

Examples

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

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.

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Beta-dual space

Definition

Examples

Properties

See also: Beta-dual space, Algebraic dual, Continuous dual, Continuous linear form, FK-AK space, Functional analysis, Linear subspace, Mathematics, Sequence space