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
![{\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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43bea6cf59e5e644006c0c8ef5317354335faa5d)
Here,
so that
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
![{\displaystyle f_{y}(x):=\sum _{i=1}^{\infty }x_{i}y_{i}\qquad x\in X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc2ac10adac9be0da5b80dc54c8ad12826bb541)
Examples
![{\displaystyle c_{0}^{\beta }=\ell ^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78eebfb60ab9893f93e2e3d9cb8dc9892e1ee6f1)
![{\displaystyle (\ell ^{1})^{\beta }=\ell ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d27c97a5a04b10a3fb0792f2a228f9aac93d86)
![{\displaystyle \omega ^{\beta }=\{0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afd5e75e2ec8ad690f5b0a4fbb38bb82897e07e1)
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.