Frame functions, signed measures and completeness of inner product spaces

Anatolij Dvurečenskij

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In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space $S$ is complete if and only if there exists a $σ$ -additive state on $C (S)$ , the orthomodular poset of complete-cocomplete subspaces of $S$ . We then consider the problem of whether every state on $E (S)$ , the class of splitting subspaces of $S$ , can be extended to a Hilbertian state on $E (\bar{S})$ ; we show that for the dense hyperplane $S$ (of a separable Hilbert space) constructed...