Strong Barrelledness Properties in Lebesgue-Bochner Spaces

If (Ω,Σ,µ) is a finite atomless measure space and X is a normed space, we prove that the space Lp(µ, X), 1 ≤ p ≤∞is a barrelled space of class ℵ0, regardless of the barrelledness of X. That enables us to obtain a localization theorem of certain mappings defined in Lp(µ, X). By “space ” we mean a “real or complex Hausdorff locally convex space”. Given a dual pair (E,F), as usual σ(E,F) denotes the weak topology on E. If B is a subset of a linear space E, 〈B 〉 will denote its linear hull. Let s be a positive integer, then a family W = {Em1m2...mp,mr ∈ N, 1 ≤ r ≤ p ≤ s} of subspaces of E is said to be an s-net in E if {Em1,m1 ∈ N} is an increasing covering of E and {Em1m2...mj,mj ∈ N} is an increasing covering of Em1m2...mj−1, for 2 ≤ j ≤ s. A space E is suprabarrelled, or barrelled space of class 1, [17], if given an increasing covering of subspaces of E there is one of them which is dense and barrelled. For a given natural number s ≥ 2, E is said to be barrelled of class s, [11] and [14], if given an increasing covering of subspaces of E there is one of them which is barrelled of class s − 1. If E is barrelled of class s for each s ∈ N it is said that E is barrelled of class ℵ0. It is shown in [11] that each non-normable Fréchet space has a dense subspace of class s − 1 which is not barrelled of class s for all s ≥ 2. On the other hand, if Σ denotes any σ-algebra of subsets of a set Ω, the space ℓ ∞ 0 (Ω, Σ) of all scalar Σ-simple functions defined on Ω with the supremum norm is a barrelled space of class ℵ0, [9]. Since each barrelled space of class s is Baire-like, [15], it Received by the editors January 199