There are excellent books on both functional analysis and summability. Most of them are very terse. In Functional Analysis and Summability, the author makes a sincere attempt for a gentle introduction of these topics to students. In the functional analysis component of the book, the Hahn–Banach theorem, Banach–Steinhaus theorem (or uniform boundedness principle), the open mapping theorem, the closed graph theorem, and the Riesz representation theorem are highlighted. In the summability component of the book, the Silverman–Toeplitz theorem, Schur’s theorem, the Steinhaus theorem, and the Steinhaus-type theorems are proved. The utility of functional analytic tools like the uniform boundedness principle to prove some results in summability theory is also pointed out.
- A gentle introduction of the topics to the students is attempted.
- Basic results of functional analysis and summability theory and their applications are highlighted.
- Many examples are provided in the text.
- Each chapter ends with useful exercises.
This book will be useful to postgraduate students, pre-research level students, and research scholars in mathematics. Students of physics and engineering will also find this book useful since topics in the book also have applications in related areas.
Table of Contents
1. Some Basic Concepts in Functional Analysis. 2. Linear Transformations, Linear Functionals and Convexity. 3. Hahn-Banach Theorem. 4. Reﬂexivity. 5. Banach-Steinhaus Theorem. 6. Closed Graph Theorem and Open Mapping Theorem. 7. Hilbert Spaces. 8. Silverman-Toeplitz Theorem and Schur’s Theorem. 9. Steinhaus Type Theorem.
P.N. Natarajan (Pinnangudi Narayanasubramanian Natarajan) is former professor and Head, Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai. He got Ph.D. in Analysis from the University of Madras in 1980. An active researcher till date, Prof. Natarajan has over 120 research papers to his credit, published in reputed journals like Proceedings of the American Mathematical Society, Journal of the London Mathematical Society, Indagationes Mathematicae, Annales Mathematiques Blaise Pascal, Commentationes Mathematicae (Prace Matematyczne), p-adic Numbers, Ultrametric Analysis and Applications and Journal of the Indian Mathematical Society. He has so far authored 5 books, published by Springer-Verlag, John Wiley and Taylor and Francis. He has also contributed a chapter each to 3 edited volumes, brought out by Springer-Verlag, Birkhauser-Verlag and Taylor and Francis. His research interests include Summability Theory and Functional Analysis, both Classical and Ultrametric. Prof. Natarajan was honoured with the Dr. Radhakrishnan Award for the Best Teacher in Mathematics for the 1990-1991 by the Government of Tamil Nadu. Besides visiting several institutes of repute in Canada, France, Holland and Greece on invitation, Prof. Natarajan has taken part in several International Conferences and has chaired sessions.