Only `free' measures are admissable on F(S) when the inner product space S is incomplete

Using elementary arguments and without having to recall the Gleason Theorem, we prove that the existence of a nonsingular measure on the lattice of orthogonally closed subspaces of an inner product space S is a sufficient (and of course, a necessary) condition for S to be a Hilbert space.

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Bibliographische Detailangaben
Veröffentlicht in:	Proceedings of the American Mathematical Society 2008-03, Vol.136 (3), p.919-922
Hauptverfasser:	Buhagiar, D., Chetcuti, E.
Format:	Artikel
Sprache:	eng
Schlagworte:	Algebra Combinatorics. Ordered structures Exact sciences and technology Functional analysis General mathematics General, history and biography Inner products Mathematical analysis Mathematical completeness Mathematical lattices Mathematical theorems Mathematics Order, lattices, ordered algebraic structures Sciences and techniques of general use
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Beschreibung
Zusammenfassung:	Using elementary arguments and without having to recall the Gleason Theorem, we prove that the existence of a nonsingular measure on the lattice of orthogonally closed subspaces of an inner product space S is a sufficient (and of course, a necessary) condition for S to be a Hilbert space.
ISSN:	0002-9939 1088-6826
DOI:	10.1090/S0002-9939-07-08982-4