In functional analysis and related areas of mathematics, the beta-dual or -dual is a certain linear subspace of the algebraic dual of a sequence space.
Definition
Given a sequence space , the -dual of is defined as
:<math>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 \}.</math>
Here, <math>\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}</math> so that <math>\mathbb{K}</math> denotes either the real or complex scalar field.
If is an FK-space then each in defines a continuous linear form on
:<math>f_y(x) := \sum_{i=1}^{\infty} x_i y_i \qquad x \in X.</math>
Examples
- <math>c_0^\beta = \ell^1</math>
- <math>(\ell^1)^\beta = \ell^\infty</math>
- <math>\omega^\beta = \{0\}</math>
Properties
The beta-dual of an FK-space is a linear subspace of the continuous dual of . If is an FK-AK space then the beta dual is linear isomorphic to the continuous dual.