On complete-cocomplete subspaces of an inner product space
In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space S is complete if and only if there exists a σ-additive state on C(S), the orthomodular poset of complete-cocomplete subspaces of S. We then consider the problem of w...
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| Veröffentlicht in: | Applications of mathematics (Prague) 2005-04, Vol.50 (2), p.103-114 |
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| creator | Buhagiar, David Chetcuti, Emanuel |
| description | In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space S is complete if and only if there exists a σ-additive state on C(S), the orthomodular poset of complete-cocomplete subspaces of S. We then consider the problem of whether every state on E(S), the class of splitting subspaces of S, can be extended to a Hilbertian state on E([bar S]); we show that for the dense hyperplane S (of a separable Hilbert space) constructed by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111-2117, every state on E(S) is a restriction of a state on E([bar S]).[PUBLICATION ABSTRACT] |
| doi_str_mv | 10.1007/s10492-005-0007-1 |
| format | Article |
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| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; SpringerNature Journals |
| subjects | Applications of mathematics Completeness Constrictions Construction Criteria Hilbert space Hyperplanes Mathematical problems Splitting Studies Subspaces |
| title | On complete-cocomplete subspaces of an inner product space |
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