Distinguished vector-valued continuous function spaces and injective tensor products (Q2080709)

Dieudonné and Schwartz called a locally convex topological vector space \(E\) distinguished if its continuous dual \(E'\) endowed with the topology of uniform convergence on all bounded subsets of \(E\) is barrelled (so that the uniform boundedness principle holds for families of operators from \(E'\) to another locally convex space \(F\)). Since Schwartz's distribution theory is based on duality for locally convex spaces, distinguishedness is a quite natural condition which is satisfied by most of the spaces appearing in distribution theory. The present article continues the authors' study of spaces \(C_p(X)\) of continuous functions on a Tychonov space \(X\) endowed with the topology of \textit{pointwise} convergence for which distinguishedness is a very restrictive property. The main results give conditions when the injective tensor product \(C_p(X)\otimes_\varepsilon E\) is distinguished, which, for example, is the case if \(X\) is countable and \(E\) is metrizable and distinguished.