Beta-dual space

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


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

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

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


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

Examples

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

Properties
The beta-dual of an FK-space $E$ 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.