Mathematical Sciences: Quasi-Banach Spaces and Their Applications

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Kalton, Casazza, and Montgomery-Smith will continue their research into the structure of Banach and non-locally convex spaces, and their applications to other areas of analysis, and applications of probability to Banach spaces. Among the questions to be addressed are whether every infinite-dimensional quasi-Banach space has a proper closed infinite-dimensional subspace, how to calculate the expected value of the norm of the sum of a Rademacher series, and whether the compact approximation property in the dual of a Banach space implies the property for the Banach space. Banach space theory is that part of mathematics that attempts to generalize to infinitely many dimensions the structure of 3-dimensional Euclidean (i.e.ordinary) space. The axioms for the distance function in a Banach space are more relaxed than those for Euclidean distance (For example, the "parallelogram law" is not required to hold.), and as a result, the "geometry" of a Banach space can be quite exotic. Much of the research in this area concerns studying the structure theory of Banach spaces.