Optimization and Differentiation in Banach Spaces

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Mamghadery, Hamid | 2011

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 43318 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ranjbar Motlagh, Alireza
Abstract:
In this thesis, we introduce and prove the inequalities and theorems of Smulian which are about derivatives of convex functions and optimization of linear functionals. Then we show that some old ideas of Smulian can be used to give another proof of theorems of Bourgain. The first theorem of Bourgain associates to any bounded, convex, closed and dentable subset like D of a Banach space X a G-delta subset of the dual space and the second theorem of Bourgain investigates the optimization in Banach spaces. Finally, we characterize subsets of Banach spaces having the Radon-Nikodym property by means of optimization results.
Keywords:
Freshet Differentiation ; Radon-Nikodym Property ; Upper Semicontinuous Maps ; Dentable Subsets ; Smulian Theorems ; Bourgain Theorems