In this project we propose to use set-theoretic methods in order to study two sets of problems coming from functional analysis. The first concerns with the problem of the continuous images of Radon-Nikodym compact spaces posed by Namioka some thirty years ago. This class of compacta has been isolated in the early 1970 and apos;s in the course of studying a Radon-Nikodym property of vector measures. Particularly useful was the characterization in terms of Frechet differentiability leading to a purely topological formulation of the problem. The other is the problem of the uniform regularity of Borel measures, posed by Pol for Rosenthal compacta and by Fremlin for general first countable compacta (it is one prominent item of his famous list of problems).
We plan to supplement the standard techniques in dealing with this kind of problems with advanced methods of set theory and Ramsey theory, and particularly the method of forcing. It is generally believed that one of the reasons why these problems have been open for so long time is the fact that so far too elementary set-theoretic methods have been applied. During the last thirty or forty years set theory has advanced considerably and specially its method of forcing invented by Paul J. Cohen some forty years ago in order to prove the independence of the Continuum Hypothesis. By now this method has been developed to the point of being able to obtain results that are not necessarily consistency results.
Moreover in recent times this method has been successfully merged with another powerful method, the Ramsey-theoretic method. We hope that the combination of these two methods is likely to give us a substantial progress on these two long- standing open problems of functional analysis.
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