Descriptive Classification Theory and Separable Banach Spaces

Three Notions of Classification Consider your favorite class ofmathematical structures, be it groups, modules, measure-preserving transformations, C-algebras, Lie groups, smooth manifolds, or something completely different. With some probability, the classification problem for these objects, that is, the problem of determining the structures up to some relevant notion of isomorphism, is, or has been, one of the central problems of the corresponding field of study. Of course, inasmuch as mathematical theories stem from attempts to model or organize physical or other phenomena, the classification problem might not be the primordial challenge. But once the basic theorems of a theory have been worked out, there is often an internal motivation to categorize its different models. For example, the definition of a Banach space as a complete normed vector space is motivated by the study of function spaces as the potential solution sets to various differential equations modeling physical phenomena. But, as is known to all of us, the common aspects of the individual problems often simplify through abstraction, whence the concept of an abstract Banach space. And therefore having an isomorphic classification of Banach spaces would certainly be helpful when dealing with more concrete problems involving these spaces.