A signed measure completeness criterion

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e. of all subspaces M for which M+M=S holds, possesses at least one nonzero completely additive signed measure or, equivalently, iff S possesses at least one nonzero frame function. Moreover, we show a new and simple proof that S is complete iff E(S) contains the join of any sequence of orthogonal one-dimensional subspaces.