We show that a formula'.x; y/ is stable if and only if ' is the pairing map on the unit ball of E£ E ⁄ , where E is a reflexive Banach space. The result remains true if the formula' is replaced by a set of formulas p.N x;N y/. In areas of mathematics where the object of study is a class of mathematical structures, one wishes to classify the structures in the class by drawing dividing lines between the simpler and the more complex structures of the class. The purpose of this paper is to point out a rather striking analogy between classification programs in two fields of mathematics which one does not normally regard as being closely related: model theory and Banach space theory. In model theory, a clear dividing line is recognized between two kinds of models: stable and unstable models. A natural measure of the complexity of a model is given by its space of types, and a stable model is one whose space of types is not much larger than the model itself. A similar distinction exists in Banach space geometry between reflexive and nonre- flexive spaces. A Banach space is reflexive if it equals its double dual. An equivalent formulation is that Banach space is reflexive if and only if its unit ball is weakly compact. Intuitively, this can be taken to mean that the unit ball of dual of the space is not much larger than the unit ball of the space itself. In 1964, R. C. James proved the following criterion for reflexivity (7). Theorem (James Condition for Reflexivity) . The following conditions are equivalent for a Banach space E. 1. E is not reflexive; 2. For everyµ 2 R with 0 <µ <1 there exists a sequence.am/ withkamkD1 for every m and a sequence. fn/ of linear functionals withk fnkD1 for every n, such that
['José Iovino']
Stable models and reflexive Banach spaces