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].