  In this paper we analyse the structure of the spaces of smooth type
functions, generated by elements of arbitrary Hilbert spaces, as a continuation
of the research in our previous papers in this series. We prove that these
spaces are perfect sequence spaces. As a consequence we describe the tensor
structure of sequential mappings on the spaces of smooth type functions and
characterise their adjoint mappings. As an application we prove the
universality of the spaces of smooth type functions on compact manifolds
without boundary.
