  In this paper we introduce and study the multiplication among smooth
functions and Schwartz families. This multiplication is fundamental in the
formulation and development of a spectral theory for Schwartz linear operators
in distribution spaces, to introduce efficiently the Schwartz eigenfamilies of
such operators and to build up a good functional calculus for them. The
definition of eigenfamily is absolutely natural and this new operation allows
us to develop a rigorous and manageable spectral theory for Quantum Mechanics,
since it appears in a form extremely similar to the current use in Physics.
