Differentiability, Summability, and Fixed Points in Banach SpacesSivek, Jeromy (2014) Differentiability, Summability, and Fixed Points in Banach Spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractThis document consists of three main chapters. Each chapter considers a topic within functional analysis. The first chapter focuses on fixed point theory. Our main result in this chapter is to show the existence of a fixed point free contractive map on a weakly compact and convex set. This answers a long-standing open question. We also prove a theorem about transfinite iterates of contractive maps on weakly compact sets converging to a fixed point. The second chapter concerns summability theory. We prove a theorem quantifying the extent to which iterated Cesaro averaging can (and cannot) bring a divergent sequence closer to convergence. We develop a method which generates Banach limits which are invariant under certain operators which generate summability methods, including the Cesaro method. We also develop a constructive method for defining transfinite iterates of a "translated" Cesaro operator corresponding to certain limit ordinals. These iterates usually have non-constructive definitions. In the third chapter we consider a couple unusual reformulations of the derivative in Banach spaces and sees what becomes of the theory of differentiation. We show that through the use of "difference transforms" one can re-work much of the foundation of Banach space differential calculus. Sometimes this re-working leads to a much more efficient development of the theory. Sometimes the results generated are different from their ordinary Frechet derivative counterparts. And sometimes, as is the case with the Inverse Function Theorem, the results and proofs are necessarily very similar to their known versions. Our main theorem is to show that in the presence of certain geometric conditions relating to smoothness, these difference transforms can be made to vary continuously in a way that is more consistent with their behavior in Hilbert spaces. We also clarify Henri Cartan's use of the term "strong derivative". Share
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