Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T23:20:10.072Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  09 April 2009

Emmanuel Chetcuti
Affiliation:
Mathematical Institute Slovak Academy of SciencesŠtefánikova 49 SK-814 73 Bratislava Slovakia e-mail: [email protected], [email protected]
Anatolij Dvurečenskij
Affiliation:
Mathematical Institute Slovak Academy of SciencesŠtefánikova 49 SK-814 73 Bratislava Slovakia e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Dorofeev, S. V. and Sherstnev, A. N., ‘Functions of frame type and their applications’, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (1990), 2329 (in Russian);Google Scholar
translation in Soviet Math. 34 (1990), 2531.Google Scholar
[2]Dvurečenskij, A., Gleason's theorem and its applications (Kluwer Acad. Publ., Dordrecht, Ister Science Press, Bratislava, 1992).Google Scholar
[3]Dvurečenskij, A. and Pták, P., ‘On states on orthogonally closed subspaces of an inner product space’, Letters Math. Phys. 62 (2002), 6370.CrossRefGoogle Scholar
[4]Gleason, A. M., ‘Measures on the closed subspaces of a Hilbert space’, J. Math. Mech. 6 (1957), 885893.Google Scholar
[5]Hamel, G., ‘Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: f (x+y) = f(x) + f (y)’, Math. Anal. 60 (1905), 459462.CrossRefGoogle Scholar
[6]Hamhalter, J. and Pták, P., ‘A completeness criterion for inner product spaces’, Bull. Londno Math. Soc. 19 (1987), 259263.CrossRefGoogle Scholar
[7]Maeda, F. and Maeda, S., Theory of symmetric lattices (Springer, Berlin, 1970).CrossRefGoogle Scholar