Measure-Theoretic Calculus in Abstract Spaces On the Playground of Infinite-Dimensional Spaces

This monograph provides a rigorous, encyclopedic treatment of the fundamental topics in real analysis, functional analysis, and measure theory.  The result of many years of the author’s careful and extensive work, this text synthesizes and builds upon the existing literature in an effort to develop and solidify the theory of measure-theoretic calculus in abstract spaces.  Standard results and proofs are illustrated in general abstract settings under rigorous treatment, and numerous ancillary topics are also covered in detail, such as functional analytic treatment of optimization, probability theory, and the theory of Sobolev spaces.  Applied mathematicians and researchers working in control theory, operations research, economics, optimization theory, and many other areas will find this text to be a comprehensive and invaluable resource.  It can also serve as an analysis textbook for graduate-level students.

- 1. Introduction. - 2. Set Theory. - 3. Topological Spaces. - 4. Metric Spaces. - 5. Compact and Locally Compact Spaces. - 6. Vector Spaces. - 7. Banach Spaces. - 8. Global Theory of Optimization. - 9. Differentiation in Banach Spaces. - 10. Local Theory of Optimization. - 11. General Measure and Integration. - 12. Differentiation and Integration. - 13. Hilbert Spaces. - 14. Probability Theory.

Zigang Pan was born in 1968 in Shanghai, China.  He received his B.S. degree in Automatic Control from Shanghai Jiao Tong University in 1990, and the M.S. and Ph.D. degrees in Electrical Engineering from the University of Illinois at Urbana-Champaign in 1992 and 1996, respectively. In 1996, he was a Research Engineer at the Center for Control Engineering and Computation at the University of California, Santa Barbara. In the same year, he joined the Polytechnic University, Brooklyn, NY, (now the Tandon School of Engineering at the New York University) as an Assistant Professor in the Department of Electrical Engineering for two years. Afterwards, he joined the faculty of Shanghai Jiao Tong University, Shanghai, P. R. China, in the Department of Automation. In January 2001, he joined the Department of Electrical and Computer Engineering and Computer Science of the University of Cincinnati as an Assistant Professor.  Since 2005, he is pursuing research on hisown interest.