Uniformly convex Banach spaces are reflexive - constructively

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the MilmanPettis theorem that uniformly convex Banach spaces are reflexive. Our aim in this note is to present a fully constructive analysis of the Milman-Pettis theorem [11, 12, 9, 13]: a uniformly convex Banach space is reflexive. First, though, we need to outline the constructive context of the statement and proof of this theorem, and to clarify the terms we will use. The context of our work is that of a quasinormed space: that is, a linear space X equipped with a family (‖ ‖i)i∈I of seminorms on X such that the subset {‖x‖i : i ∈ I} of R is bounded; that family is called the quasinorm on X. For all x, x′ in the quasinormed space X we define the inequality and equality relations by: • (x 66= x′) ≡ ∃i∈I (‖x− x‖i > 0); and • (x = x′) ≡ ∀i∈I (‖x− x‖i = 0) . Quasinorms were introduced by Johns and Gibson [8] in their study of Orlicz spaces (such as L∞), and are important in the constructive analysis of duality; see [4] (Chapter 7, Section 5). An element x of the quasinormed space X is normable, or normed, if its norm, ‖x‖ ≡ sup i∈I ‖x‖i , (1) exists. The unit ball of a quasinormed space ( X, (‖ ‖i)i∈I ) is the set BX ≡ {x ∈ X : ∀i∈I (‖x‖i 6 1)} . If every element of a quasinormed space ( X, (‖ ‖i)i∈I ) is normable, then we can regard X as a normed space relative to the norm defined at (1). 1Our view of constructive mathematics is that of Bishop [3, 4, 5]: namely and roughly, it is mathematics with intuitionistic logic together with some appropropriate foundation such as Aczel’s constructive (ZF) set theory [1, 2]. Note, for the record, that we accept the axiom of dependent choice. 2Those authors used the name ‘pseudonorm’, rather than our ’quasinorm’.