Recent Advances in Nonlinear Partial Differential Equations:
A Celebration of Norman Dancer’s 60th Birthday
University of New England (Armidale), Australia, 16-21 July 2006
Uniform homeomorphisms—that is to say, bijections which are uniformly continuous in both directions—induce group structures on complete normed vector spaces (Banach spaces) that “resemble” the additive structures of the underlying spaces.
Analysis of uniform Banach groups sheds new light on open questions about the classification of Banach spaces up to uniform homeomorphisms. For example, in this talk, we will detail a very general answer to a question recently highlighted by Yoav Benjamini and Joram Lindenstrauss in their monograph Geometric Nonlinear Functional Analysis (Volume 1): Can a non-normable topological vector space be uniformly homeomorphic to a Banach space?
Our approach to this question, via uniform Banach groups, determines a new linearization procedure for a wide class of uniform homeomorphisms. Such techniques are rare because uniformly continuous maps do not have derivatives in general, rendering affine approximation difficult.