Abstract
Using Soler's result, we show that the existenceof at least one finitely additive probability measure onthe system of all orthogonally closed subspaces of Swhich is concentrated on a one-dimensional subspace of E can imply that E is a real,complex, or quaternionic Hilbert space. In addition,using the concept of test spaces of Foulis and Randalland introducing various systems of subspaces of E , we give some characterizations of inner productspaces which imply that E is a real, complex, orquaternionic Hilbert space.
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Dvurecenskij, A. Soler's Theorem and Characterization of Inner Product Spaces. International Journal of Theoretical Physics 37, 23–29 (1998). https://doi.org/10.1023/A:1026600919901
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DOI: https://doi.org/10.1023/A:1026600919901