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

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Given a sequence space X, the β-dual of X is defined as

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

Examples

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Properties

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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.