Trisha Shetty (Editor)

Beta dual space

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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.

Contents

Definition

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

X β := { x ∈ X   :   ∑ i = 1 ∞ x i y i < ∞ ∀ y ∈ X } .

If X is an FK-space then each y in Xβ defines a continuous linear form on X

f y ( x ) := ∑ i = 1 ∞ x i y i x ∈ X .

Examples

  • c 0 β = ℓ 1
  • ( ℓ 1 ) β = ℓ ∞
  • ω β = ∅

    • 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.

    References

    Beta-dual space Wikipedia
    (Text) CC BY-SA