Dual systems of inductive-projective limits of Hilbert spaces originating from self-adjoint operators

In this paper we construct spaces SF( ) and TF( ) where F denotes a suitable directed set of Borel functions on n, and where denotes an n-tuple of strongly commuting self-adjoint operators. The spaces SF( ) and TF( ) are in duality. We give conditions on the set F such that SF( ) and TF( ) can be described both as a (non-strict) inductive limit and as a projective limit of Hilbert spaces. Examples are included