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 β := { x ∈ K N : ∑ i = 1 ∞ x i y i converges ∀ y ∈ X } . {\displaystyle 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, K ∈ { R , C } {\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}} so that K {\displaystyle \mathbb {K} } denotes either the real or complex scalar field. If X is an FK-space then each y in Xβ defines a continuous linear form on X
f y ( x ) := ∑ i = 1 ∞ x i y i x ∈ X . {\displaystyle f_{y}(x):=\sum _{i=1}^{\infty }x_{i}y_{i}\qquad x\in X.}
Examples
c 0 β = ℓ 1 {\displaystyle c_{0}^{\beta }=\ell ^{1}}
( ℓ 1 ) β = ℓ ∞ {\displaystyle (\ell ^{1})^{\beta }=\ell ^{\infty }}
ω β = { 0 } {\displaystyle \omega ^{\beta }=\{0\}}
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.
References