×

Completeness of inner product spaces with respect to splitting subspaces. (English) Zbl 0601.46026

Three distinguished families of subspaces of a, non necessarily complete, inner product space S are singled out from the ordering point of view.
(a) The set \(\Sigma(S)\) of all subspaces of S, whch is a weak orthocomplemented complete lattice with respect to set theoretical inclusion and the annihilation mapping.
(b) The set \(F(S)\) of all exact or \(\perp\)-closed subspaces of \(S\), which is an orthocomplemented complete lattice.
(c) The set of all splitting subspaces \(E(S)\) of \(S\), which is an orthocomplemented orthomodular orthoposet.
A standard result of inner product space theory is the theorem which ensures that S is complete
i) iff the weak orthocomplemented complete lattice \(\Sigma(S)\) is orthocomplemented and orthomodular;
ii) iff the orthocomplemented complete lattice \(F(S)\) is orthomodular;
iii) iff the orthocomplemented orthomodular orthoposet \(E(S)\) is a complete lattice.
The main result of this work is that S is a Hilbert space under the weaker request that \(E(S)\) is a \(\sigma\)-lattice. As a marginal result it is also proved that an inner product space is complete if and only if the complete lattice \(\Sigma(S)\) is orthomodular.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
06C15 Complemented lattices, orthocomplemented lattices and posets
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] AmemiyaI. and ArakiH., ?A Remark on Piron’s Paper?, Publ. Res. Inst. Math. Sci. Ser. A 12, 423-427 (1966-67).
[2] Cattaneo, G. and Marino, G., ?Non Standard Orthocomplementations on Posets and BZ-Posets?, Preprint DMUC/July (1984).
[3] GrossH. and KellerA., ?On the Definition of Hilbert Space?, Manuscripta Math. 23, 67-90 (1977). · Zbl 0365.46023
[4] GudderS. P., ?Axiomatic Quantum Mechanics and Generalized Probability Theory?, in A.Bharucha-Reid (ed.), Probabilistic Methods in Applied Mathematics, Vol. II, Academic Press, New York, 1970.
[5] GudderS. P., ?Inner Product Spaces?, Am. Math. Monthly 81, 29-36 (1974). · Zbl 0279.46013
[6] GudderS. P., Correction to ?Inner Product Spaces?, Am. Math. Monthly 82, 251-252 (1975). · Zbl 0305.46033
[7] GudderS. P. and HollandS., Second correction to ?Inner Product Spaces?, Am. Math. Monthly, 82, 818 (1975). · Zbl 0305.46033
[8] PiziakR. ?Orthomodular Posets from Sesquilinear Forms?, J. Australian Math. Soc. 15, 265-269 (1973). · Zbl 0271.15013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.