A Characterization of Reflexivity

We give a characterization of reflexivity in terms of rotundity of the norm. Renorming characterization of various classes of Banach spaces is important and useful for applications. Some classes turn out to have very elegant descriptions, while most seem to resist the renorming point of view. The most spectacular result in this area is certainly the Enflo-Pisier characterization of superreflexive spaces as those admitting uniformly rotund norm [E] (or even having power type modulus of uniform convexity [P]). Restricting to separable Banach spaces allows more elegant results (not valid in the general case), such as Fréchet smooth or weakly uniformly rotund (WUR) renorming characterization of spaces with a separable dual (Asplund spaces), characterizations of subspaces of c0 etc. A good source of references on the subject is G. Godefroy’s article in [JL], or [DGZ]. In the present note we are interested in renorming characterization of reflexivity. Let us give a brief account of the known facts. Combining Theorem 5.4 of [C] with the fundamental LUR renorming of the WCG spaces [T], Troyanski obtained a characterization of reflexive spaces X as those admitting a renorming ‖ · ‖ with the following property (named weakly 2-rotund (W2R) by Cudia [C]): For every sequence {xn} ⊂ SX , if there exists 0 6= f ∈ X ∗ satisfying lim m,n→∞ |f(nm 2 )| = ‖f‖, then {xn} is convergent in norm. We can easily see that this happens if and only if each f ∈ SX∗ strongly exposes the unit ball of X. It is standard (using Šmulyan’s criterion) and well-known that the above property of the norm ‖ · ‖ is equivalent to ‖·‖ being Fréchet smooth. In particular, every LUR renorming of a reflexive space satisfies the criterion. (Note however, that in [T] Cudia’s definition of W2R is stated incorrectly, and in fact the stated condition fails to imply reflexivity as shown in [HR].) On the other hand, Milman in [M] introduced the notions of 2-rotund (2R) and weakly 2-rotund (W2R). (See below for these definitions and note that Milman’s W2R is distinct from Cudia’s. In the present paper we choose to use Milman’s terminology which seems more in place.). He states (without proof) that separable reflexive spaces are precisely the W2R renormable and asks whether reflexive spaces with LUR norm (this condition is redundant due to [T]) are 2R renormable. The last problem was settled positively for separable spaces by Odell and Schlumprecht [OS], but the general case remains open. The main result of this note is a characterization of reflexive spaces as those admitting a W2R renorming. We also give examples showing that LUR renorming of a reflexive space is not necessarily W2R and vice versa, so ours is an essentially different characterization of reflexivity than Cudia-Troyanski’s. First let us fix some notation. For a finite set A we denote the number of elements of A by |A|. Given a vector x = {x(γ)}Γ ∈ c0(Γ) and an A ⊂ Γ, x↾A denotes the vector defined as x↾A (γ) = x(γ) for γ ∈ A and x↾A (γ) = 0 for γ ∈ Γ \A. 1991 Mathematics Subject Classification. 46B20, 46B03, 46B10.