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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
\( X^{\beta}:= \left \{ x \in X \ : \ \sum_{i=1}^{\infty} x_i y_i < \infty \quad \forall y \in X \right \} \).
If X is an FK-space then each y in Xβ defines a continuous linear form on X
\( f_y(x) := \sum_{i=1}^{\infty} x_i y_i \qquad x \in X. \)
Examples
\(c_0^\beta = \ell^1\)
\( (\ell^1)^\beta = \ell^\infty\)
\( \omega^\beta = \emptyset\)
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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