Beta-dual space

Beta-dual space

In functional analysis and related areas of mathematics, the beta-dual or eta-dual is a certain linear subspace of the algebraic dual of a sequence space.

Definition

Given a sequence space X the eta-dual of X is defined as:X^{eta}:={x in omega : sum_{i=1}^{infty} x_i y_i < infty quad forall y in X}.

If X is an FK-space then each y in X^{eta} 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^eta = l^1
* (l^1)^eta = l^infty
* omega^eta = Phi

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.


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