One of the main problems in the theory of inductive limits of Banach
spaces is the
projective description problem, finding a reasonable representation for
the
continuous seminorms. The problem is nontrivial even in the simplest cases.
Recall
that given, for example, an increasing sequence of Banach spaces
(Yk)∞k=1
with
continuous embeddings Yk[rarrhk ]Yk+1
the inductive limit is the space
Y=∪kYk endowed
with the finest locally convex topology τ such that every embedding
Yk[rarrhk ](Y, τ)
becomes continuous. It is possible to give abstract definitions for families
of
continuous seminorms generating the topology τ, but the connection
with the norms
of the step spaces Yk is not necessarily
very close. For example, if the spaces Yk
are
Banach spaces of continuous functions endowed with weighted sup-norms,
it is not
clear if the continuous seminorms of the inductive limit are of the same
type.
We mention that inductive limits of spaces of continuous and holomorphic
functions occur in many areas of analysis like linear partial differential
operators,
convolution equations [BD1], [E],
complex and Fourier analysis and distribution
theory. The projective description problem in these spaces has been thoroughly
studied in [BMS1, BB1, BB2, BB3, BT, BM1, BM2], to mention
some examples. We
refer to the survey articles [BM1,BMS2, BB3].
The present work is also connected
with the factorization problems which are treated in the book [Ju].