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Boundedness of sign-preserving charges, regularity, and the completeness of inner product spaces

Published online by Cambridge University Press:  09 April 2009

Emmanuel Chetcuti  and
Anatolij Dvurečenskij
Emmanuel Chetcuti
Affiliation:
Mathematical Institute Slovak Academy of SciencesŠtefánikova 49 SK-814 73 Bratislava Slovakia e-mail: chetcuti@mat.savba.sk, dvurecen@mat.savba.sk
Anatolij Dvurečenskij
Affiliation:
Mathematical Institute Slovak Academy of SciencesŠtefánikova 49 SK-814 73 Bratislava Slovakia e-mail: chetcuti@mat.savba.sk, dvurecen@mat.savba.sk
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Abstract

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We introduce sign-preserving charges on the system of all orthogonally closed subspaces, F(S), of an inner product space S, and we show that it is always bounded on all the finite-dimensional subspaces whenever dim S = ∞. When S is finite-dimensional this is not true. This fact is used for a new completeness criterion showing that S is complete whenever F(S) admits at least one non-zero sign-preserving regular charge. In particular, every such charge is always completely additive.

Keywords

Inner product spaceorthogonally closed subspacechargesign-preserving chargeregular chargebounded chargecompleteness of inner product space

MSC classification

Secondary: 81P10: Logical foundations of quantum mechanics; quantum logic
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 2 , April 2005 , pp. 199 - 210
Copyright
Copyright © Australian Mathematical Society 2005

References

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