Professor Alspach's part of this project will continue his investigation of the complemented subspaces of spaces of continuous functions and of functions whose p-th powers are integrable. The long-term goal is to classify up to isomorphism the complemented subspaces of such spaces. Professor Schutt will study finite-dimensional subspaces of the space of integrable functions and investigate certain features of the convex geometry of Banach spaces. 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.
