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Completeness of inner product spaces with respect to splitting subspaces

✍ Scribed by Gianpiero Cattaneo; Giuseppe Marino

Publisher
Springer
Year
1986
Tongue
English
Weight
222 KB
Volume
11
Category
Article
ISSN
0377-9017

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✦ Synopsis

The set E(S) of all splitting subspaces, i.e., of all subspaces M of an inner product space S for which M(~ M β€’ = S holds, is an orthocomplemented orthomodular orthoposet and it has already been shown that the ordering property on E(S) of being a complete lattice characterizes the completeness of inner product spaces. In this work this last result is generalized proving that S is a Hilbert space under the weaker request that E(S) is a a-lattice. As a marginal result, we also prove that an inner product space is complete if and only if the complete lattice Z(S) of all subspaces is orthomodular.

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