Mathematical Sciences: Structure of Infinite-Dimensional Banach Spaces

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Professors Rosenthal and Odell will work on a variety of problems in the theory of Banach spaces. One central problem concerns identifying the complemented subspaces of the space of absolutely integrable functions on an interval. This is related to the structure of Banach spaces and convex sets failing the Radon-Nikodym property. The question of the existence of subspaces isomorphic to the space of null or absolutely summable sequences in a hereditarily non-reflexive Banach space will be pursued. The approach Rosenthal and Odell will use involves analysis of the structure of spaces of Baire functions. Their recently invented notion of governance of a class of Banach spaces by a function will also play an important role. The theory of Banach spaces is about analysis in infinitely many dimensions. For each finite dimension n, there is only one n-dimensional space, in most respects as well understood in general as in the familiar cases when n is 1, 2, or 3. These are used to keep track of situations in which there are only finitely many degrees of freedom. When the number of degrees of freedom becomes infinite, as would for instance be necessary to describe the configuration of a piece of string, there are many possible reasonable choices for the ambient space in which to perform analysis. The work supported by this award is largely concerned with the ways in which these spaces fit into another, a line of investigation that lays bare a good deal of their structure.