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| In [[functional analysis]] and related areas of [[mathematics]], the '''beta-dual''' or '''<math>\beta</math>-dual''' is a certain linear subspace of the [[algebraic dual]] of a [[sequence space]].
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| == Definition ==
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| Given a sequence space <math>X</math> the '''<math>\beta</math>-dual''' of <math>X</math> is defined as
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| :<math>X^{\beta}:=\{x \in X \mid \sum_{i=1}^{\infty} x_i y_i < \infty \quad \forall y \in X\}.</math>
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| If <math>X</math> is an [[FK-space]] then each <math>y</math> in <math>X^{\beta}</math> defines a [[continuous linear form]] on <math>X</math>
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| :<math>f_y(x) := \sum_{i=1}^{\infty} x_i y_i \qquad x \in X.</math> | |
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| == Examples ==
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| * <math>c_0^\beta = l^1</math>
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| * <math>(l^1)^\beta = l^\infty</math>
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| * <math>\omega^\beta = \emptyset</math>
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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.
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| {{mathanalysis-stub}}
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| [[Category:Functional analysis]]
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Latest revision as of 04:11, 26 July 2014
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