The Uniform Classification of Banach Spaces

This is a survey of results on the classification of Banach spaces as metric spaces. It is based on a series of lectures I gave at the Functional Analysis Seminar in 1984-1985, and it appeared in the 1984-1985 issue of the Longhorn Notes. I keep receiving requests for copies, because some of the material here does not appear elsewhere and because the Longhorn Notes are not so easy to get. Having it posted on the Bulletin thus seems reasonable despite the fact that it is not updated, and I thank the Editors of the Longhorn Notes for their permission to do so. §0. Introduction. Banach spaces are topological (metric) spaces with an additional structure — the vector space structure. In the linear theory we study all these structures simultaneously, and we deal with linear continuous maps. Two spaces are identified if they are linearly homeomorphic. One could, however, consider Banach spaces as a special class of topological (or metric) spaces and study them as such. In the topological classification two spaces are identified if they are homeomorphic. In the metric classification we identify uniformly-homeomorphic spaces. While the linear theory is very rich — there are many different types of spaces, with complicated subspace structure, the topological theory is, in some sense, trivial. A remarkable theorem of Kadec says that any two separable Banach spaces are homeomorphic. (See [BP] for a thorough study of Banach spaces as topological spaces.) Kadec’s theorem was extended by Torunczyk [T], who proved that two Banach spaces are homeomorphic iff they have the same density character. The uniform theory lies between these two extremes. It is rich enough so that to say that two Banach spaces are uniformly homeomorphic already says something about similarities in their linear structure. Yet it does not, in general, imply linear isomorphism. In this series of lectures I presented some of the ideas and results in the theory of uniform classification. There is no attempt at a comprehensive survey. I chose one topic — the infinite dimensional classification problem, and presented, with Partially supported by NSF Grant DMS 8403669. Typeset by AMS-TEX 1