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Ordering on families of subspaces of pre-Hilbert spaces and Dacey pre- Hilbert spaces. (English) Zbl 0637.46025

Let S denote a pre-Hilbert space which is not complete in general. If an orthonormal set \(\{U_{\alpha}\}\) in S, ons for short, satisfies the condition such that \(\{U_{\alpha}\}\subseteq \{x\}^{\bot}\cup \{y\}^{\bot}\) leads to \(x\bot y\), \(\{U_{\alpha}\}\) is said to be Dacey. If and only if any maximal orthonormal system is Dacey, S is said to be a Dacey pre-Hilbert space. When ons satisfies Parseval’s equality, it is said to be an orthonormal basis. Then, any orthonormal basis is Dacey, and any Dacey orthonormal system is maximal. In S there is a dissymmetric behaviour between the set of all subspaces and the set of all orthogonal projections from the ordering point of view. The authors try to show it by using the following six distinct families of subspaces M of S:
\(\sum (S)=\{M:\) M subspace}; \(F(S)=\{M:M=M^{\bot \bot}\}\); \(D(S)=\{M:^{\exists}ons\{U_{\alpha}\}\) such that \(M=\{U_{\alpha}\}^{\bot \bot}\}\); \(R(S)=\{M:M=\{U_{\alpha}\}^{\bot \bot}\) for all max ons \(\{U_{\alpha}\}\) in \(M\}\) ; \(V(S)=\{M:M,M^{\bot}\in R(S)\), \(M^{\vee}M^{\bot}=S\}\); \(E(S)=\{M:M\oplus M^{\bot}=S\}.\)
E(S) is isomorphic to the partially ordered set of orthogonal projections of S. For all \(M\in D(S)\) there exists one local complement \(M'\). Namely there exists \(M'\in D(S)\) satisfying \(M\bot M'\) and \(M^{\vee}M'=S\). S is Dacey if and only if for every \(M\in D(S)\) there exists a unique \(M'\in D(S)\) satisfying the further condition \(M'=M^{\bot}\). Their four results and one conjecture are the following:
(I) \(V(S)=R(S)=D(S)\) if and only if S is Dacey.
(II) If S is non complete and Dacey, \(D(S)\neq F(S)\) necessarily holds.
(III) If S is separable and non complete, \(R(S)\neq D(S)\) holds.
(IV) Under the hypothesis of separability, S is Dacey if and only if it is a Hilbert space.
Conjecture: \(V(S)\) is coinciding with \(E(S)\) from the ordering point of view.
Reviewer: H.Yamagata

MSC:

46C99 Inner product spaces and their generalizations, Hilbert spaces
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