Definitions

# Beta-dual space

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

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

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

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

## Examples

• $c_0^beta = l^1$
• $\left(l^1\right)^beta = l^infty$
• $omega^beta = Phi$

## 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.

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