Bornological spaces of type \(C(X)\otimes _{\epsilon}E\) or C(X;E) (Q1089580)

Conditions are given on a compact subset K of a completely regular Hausdorff space X and on a locally convex space E to ensure that the restriction mapping \(R_ K:C(X,E)\to C(K,E)\) is a surjective homomorphism such that the bounded sets can be lifted. In general it seems to be still open whether the space C(X,E), endowed with the compact open topology is bornological. The results mentioned above are used to provide new conditions to ensure that the answer is positive. Complementary information can be seen in the article DOGA Tr. J. Math. 10, 1, 83-90 (1986) by the same authors. The related question of the commutability of inductive limits and injective tensor products with spaces of type C(X) is also considered.