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.


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.


  • c_0^\beta = \ell^1
  • (\ell^1)^\beta = \ell^\infty
  • \omega^\beta = \emptyset


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.