  In this paper we describe several new aspects of the foundations of the
representation theory of the space of smooth-automorphic forms (i.e., not
necessarily $K_\infty$-finite automorphic forms) for general connected
reductive groups over number fields. Our role model for this space of
smooth-automorphic forms is a ''smooth version'' of the space of automorphic
forms, whose internal structure was the topic of a famous paper of Franke. We
prove that the important decomposition along the parabolic support, and the
even finer - and structurally more important - decomposition along the cuspidal
support of automorphic forms transfer in a topologized version to the larger
setting of smooth-automorphic forms. In this way, we establish
smooth-automorphic versions of the main results of a paper of Franke-Schwermer
and of Moeglin-Waldspurger's book, III.2.6.
