Stable theories in functional analysis

Model theory is the logical analysis of mathematical structures. The class of structures considered in model theory includes, among others, all structures from algebra, number theory, and finite-dimensional analysis. A limitation, however, is that this class does not include the families of structures that are studied functional analysis, e.g., Banach spaces, or operator algebras. This limitation lies in the fact that the discourse in analysis is carried out almost entirely in higher order logic, and the traditional logic of model theory is first order logic. Higher order logics do not have a powerful model theory.Recently, C. W. Henson has developed a model-theoretical apparatus that extends model theory to include important classes of structures from functional analysis preserving the desirable characteristics of first-order logic. One has, for instance, a compactness theorem and a Lowenheim-Skolem theorems for Banach spaces.Some of the most remarkable results of Banach space theory in the last two decades have been proved by borrowing ideas from model theory (e.g., ultrapowers, stability). The apparatus described above provides a foundation for such contributions.We have developed Henson's framework further, and introduced the fundamentals of classification theory (or stability theory) for structures based on Banach spaces.U of I OnlyETDs are only available to UIUC Users without author permissio