Mathematical Sciences: The Geometry of Banach Spaces

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Girardi will investigate the internal geometry of Banach spaces and applications to other areas of analysis. She will concentrate on the complete continuity property and its relation to other geometric properties (CCP). She will also examine the relations between CCP, strong regularity, and the Krien-Milman property. She will also continue her investigation of the Bochner-Lebesgue 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.