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A note on homomorphisms of inner product spaces

Pulmannova S.

Demonstratio Mathematica

2005

Vol. 38, nr 4

997--1004

Recently, Buhagiar and Chetcuti [1] have shown that if V1 and V2 are two separable, real inner product spaces such that the modular ortholattices of their finite and cofmite subspaces are algebraically isomorphic, then V1 and V2 are isomorphic as inner product spaces. Their proof is based on the properties of inner product spaces, in particular it makes use of Gleason's theorem. In this note we show, using techniques of projective geometry, that their result holds for any inner product spaces, real, complex or quaternionic, of dimension at least three, not necessarily separable. We also consider the case when the algebraic isomorphism is replaced by a homomorphism, and the case when the underlying fields of V1 and V2 are not the same.