Mathematical Sciences: Functional Analysis

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DMS Proposal #9622841 PI: Dr. Maria Girardi University of South Carolina @ Columbia Girardi's ultimate goal is to gain a better understanding of Functional Analysis; her approach is to examine the interplay between its various sub-areas; at this time, her main thrust is in the branch of Banach Space Theory. She will continue her work, in which she has made progress over the past years, on several (related) long-standing problems in Banach space theory that are centered around the Radon-Nikodym Property. She will also continue examining the Bochner-Lebesgue space and begin a new project that ties together Functional Analysis and Homological Algebra. Roughly speaking, a Banach space is a collection of points that has, among other things, a way to measure the "distance" between any two points. An example is the 3-D Euclidean (ordinary) space. Another example is the space of "suitably nice" functions (so a point in this type of Banach space is a function) with distance suitably clarified. A practical application of Banach space theory is compact disc coding; a sound signal is, roughly speaking, a function in a suitable Banach space. Unfortunately the sound signal contains too much information to store on the CD so the signal is approximated by an element from the Banach space that is close to the true element (this is where the notion of distance enters). Understanding the structure of Banach spaces aids in the above, and other, practical applications.