(LB)-spaces of vector-valued continuous functions - Frerick, Leonhard; Wengenroth, Jochen (The Bulletin of the London Mathematical Society, 2008)
We consider a question posed by Bierstedt and Schmets as to whether, for an (LB)-space E and a compact space K, the space C(K, E) of E-valued continuous functions endowed with the uniform topology is again an (LB)-space. We present proofs for the case of hilbertizable Montel (LB)-spaces as well as for the case where E is a weighted space of sequences or functions which even yield that Schwartz's ε-product Y ε E is an (LB)-space for every -space Y. Although the problem for general (LB)-spaces remains open, we provide several relations and reductions. For instance, it is enough to consider curves, that is, the most natural case K=[0, 1] or, for the class of hilbertizable (LB)-spaces with metrizable bounded sets, the Stone–Čech compactification of ℕ.