Abstract
For any unit vector in an inner product space S, we define a mapping on the system of all ⊥-closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. Finally, the result will be generalized to general inner product spaces.
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Dvurečenskij, A., Neubrunn, T. & Pulmannová, S. Finitely additive states and completeness of inner product spaces. Found Phys 20, 1091–1102 (1990). https://doi.org/10.1007/BF00731854
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DOI: https://doi.org/10.1007/BF00731854