A Completeness Criterion for Inner Product Spaces
Jan Hamhalter, Pavel Pták · Bulletin of the London Mathematical Society · 1987
We show that a separable inner product space is complete if and only if its lattice of strongly closed subspaces possesses at least one state. This gives a measure-theoretic characterization of Hilbert spaces among inner product spaces and, as a by-product, exhibits a ‘continuous’ example of a stateless orthocomplemented lattice.