Barrelledness conditions onC 0 (E)

Some conditions of barrelledness are considered and studied on the spaceC ₀(E), defined as follows: IfE is a real or complex Hausdorff locally convex space and \(P_E \) is a saturated family of seminorms, defining the original topology ofE, then the vector space of all the sequences \(\bar f = \left\{ {\bar f(n): n \in \mathbb{N}} \right\}\) inE, convergent to zero, provided with the locally convex topology $$\bar p(\bar f) = sup\left\{ {p (\bar f(n)): n \in \mathbb{N}} \right\}p \in P_E $$ is defined as the spaceC ₀(E). The main result of the paper is the following characterization:C ₀(E) is quasibarrelled (see 3], p. 367) if and only if,E is quasibarrelled and the strong dual ofE has property (B) (see 5], p. 30, for definition). We obtain. as a consequence, commutativity properties of the operatorC ₀, acting on certain inductive limits (3.3 Theorem). We also prove thatC ₀ does not commute with uncountably strict inductive limits. In particular, there are ultrabornological spacesE for whichC ₀(E) is not quasibarrelled. 3.1. Example provides a complete?-tensor product of two complete ultrabornological spaces which is not quasibarrelled.