On the Barrelledness of the Vector-Valued Bounded Function Space

If Ω is a set, Σ a σ-algebra of subsets of Ω, and X a normed space, we show that the space l∞(Σ, X) of all bounded X-valued Σ-measurable functions defined on Ω, provided with the supremum-norm, is barrelled if and only if X is barrelled. Assuming X separable, this implies that the space l∞(Ω, X) of all bounded X-valued functions defined on Ω, endowed with the supremum-norm, is barrelled whenever X is barrelled.