On isomorphisms of inner product spaces

Abstract

In this paper, we show that if Sx and S2 are two separable, real inner product spaces such tha t P(SX) is algebraically isomorphic to P(S2), where P(S) denotes the modular lattice of finite and cofinite dimensional subspaces of an inner product space S, then Sx and S2 are isomorphic as inner product spaces. The proof makes use of Gleason's theorem. We also remark that, as a consequence of this, if for two separable, real inner product spaces S1, and S2, the respective complete lattices of strongly closed subspaces are isomorphic, then Sx and S2 are unitarily equivalent. In particular, if we just restrict ourselves to complete inner product spaces, we obtain the classical Wigner's theorem.