Beta-dual space

In functional analysis and related areas of mathematics, the beta-dual or {{math|β}}-dual is a certain linear subspace of the algebraic dual of a sequence space.

Definition

Given a sequence space {{mvar|X}}, the {{math|β}}-dual of {{mvar|X}} is defined as

: $X^{\beta}:= \left \{ x \in\mathbb{K}^\mathbb{N}\ : \ \sum_{i=1}^{\infty} x_i y_i\text{ converges }\quad \forall y \in X \right \}.$

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

If {{mvar|X}} is an FK-space then each {{mvar|y}} in {{math|X^β}} defines a continuous linear form on {{mvar|X}}

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

Examples

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

Properties

The beta-dual of an FK-space {{mvar|E}} is a linear subspace of the continuous dual of {{mvar|E}}. If {{mvar|E}} is an FK-AK space then the beta dual is linear isomorphic to the continuous dual.

Category:Functional analysis