A signed measure completeness criterion (Q910663)

Let S be a real or a complex inner product space. Let E(S) be the set of all subspaces M of S for which the condition: \(M+M^{\perp}=S\) holds. In this paper the authors show that S is complete iff E(S) possesses at least one nonzero completely additive signed measure on E(S) or, equivalently, iff S possesses at least one nonzero frame function. This main result is applying, by the authors for another system of closed subspaces [see: Lett. Math. Phys. 17, 19-24 (1988; review above)].