Test spaces,dacey spaces,and completeness of inner product spaces

We show that a test space consisting of the unit sphere of a real or complex inner product spaceS and of the set all maximal orthonormal systems inS, is algebraic iffS is Dacey or, equivalently, iffS is complete. In addition, we present another completeness criterion.This research is supported by the grant G-368 of the Slovak Academy of Sciences, Slovakia.