This note investigates, for
certain locally convex spaces which have bases and are strict inductive or projective
limits, the structural property of being a direct sum or a product. Our approach is
based on a suitably more general version of a decomposition lemma originally due to
S. Dineen and gives a better understanding of the non-existence of bases in certain
nuclear (F)- and strict (LF)-spaces. Our method also allows us to investigate the
structure of various other non-nuclear spaces with unconditional bases yielding, in
particular, examples of spaces with no such bases. In part, this also motivated us to
include some rather general remarks on the problem of when the strong dual
of a homomorphism between locally convex spaces is a homomorphism as
well.
