Completeness of inner product spaces and quantum logic of splitting subspaces
type: Item
instance of: scholarly article
title: Completeness of inner product spaces and quantum logic of splitting subspaces
zbMATH Open document ID: 0652.46017
DOI: 10.1007/BF00398592
author: Anatolij Dvurečinskij
published in: Letters in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: Let S be a real or complex inner product space and let E(S) be the set of all splitting subspaces of S, i.e., of all subspaces M of S for which the condition \(M\oplus M^{\perp}=S\) holds. In the present paper the author shows that S is a complete space under the weaker request that E(S) is a quantum logic, that is, an orthocomplemented orthomodular \(\sigma\)-orthoposed, i.e., if \(\{M_ n\}^{\infty}_{n=1}\) is a sequence of mutually orthogonal subspaces of E(S), then \(\bigvee^{\infty}_{n=1}M_ n\) exists in E(S). This result generalizes the result of G. Cattaneo and G. Marino who required for E(S) to be a lattice.
Mathematics Subject Classification ID: 46C99
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 4063793
zbMATH Keywords: inner product space
zbMATH Keywords: splitting subspaces
zbMATH Keywords: quantum logic
zbMATH Keywords: orthocomplemented orthomodular \(\sigma\)-orthoposed
reviewed by: Neculai Papaghiuc
MaRDI profile type: Publication
cites work: Completeness of inner product spaces with respect to splitting subspaces
cites work: On the definition of Hilbert space
cites work: Inner Product Spaces
cites work: Correction to: "Inner Product Spaces"
cites work: A Completeness Criterion for Inner Product Spaces
cites work: Partial Solution to Mackey's Problem about Modular Pairs and Completeness
cites work: Probability in physics and a theorem on simultaneous observability
full work available at URL: https://doi.org/10.1007/bf00398592
OpenAlex ID: W2079169496
description: scientific article; zbMATH DE number 4063793