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Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and $\sigma $-additive state
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 $\sigma $-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 by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on $E(S)$ is a restriction of a state on $E(\bar{S})$.
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