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Frame functions, signed measures and completeness of inner product spaces. (English) Zbl 0715.46031

Summary: We show that an inner product space is complete iff it possesses at least one nonzero frame function, or, equivalently, when and only when some systems of closed subspaces possess at least one nonzero signed measure.

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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