Some barrelledness properties of c0(Ω,X)

Assuming that Ω is a nonempty set and X is a normed space, we show that the real or complex linear space c0(Ω,X) of all functions f :Ω →X such that for each >0 the set {ω ∈ Ω: f (ω) > } is finite, equipped with the supremum norm, is either barrelled, ultrabornological or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological or unordered Baire-like.  2002 Elsevier Science (USA). All rights reserved.