This book is aimed to serve as a textbook for an introductory course in metric spaces for undergraduate or graduate students. It has been our goal to present the basics of metric spaces in a natural and intuitive way and encourage the students to think geometrically while actively participating in the learning of this subject. In this book, we have illustrated the strategy of the proofs of various theorems that motivate the readers to complete it on their own. Bits of pertinent history are infused in the text, including brief biographies of some of the central players in the development of metric spaces. The textbook is decomposed in to seven chapters which contain the main materials on metric spaces; namely, introductory concepts, completeness, compactness, connectedness, continuous functions and metric fixed point theorems with applications.
Some of the noteworthy features of this book are:
· Diagrammatic illustrations that encourage the reader to think geometrically.
· Focus on systematic strategy to generate ideas for the proofs of theorems.
· A wealth of remarks, observations along with variety of exercises.
· Historical notes and brief biographies appearing throughout the text.
Table of Contents
Set Theory. Sets. Relations. Functions. Countability of Sets. Metric Spaces. Review of Real Number System and Absolute Value. Young, Holder and Minkowski Inequalities.Notion of Metric Space. Open Sets. Closed Sets. Interior, Exterior and Boundary Points. Limit and Cluster Points. Bounded Sets. Distance between sets. Equivalent Metrics. Complete Metric Spaces. Sequences. Convergence of sequence. Complete Metric Spaces. Completion of metric spaces. Baire Category Theorerm. Compact Metric Spaces. Open Cover and Compact Sets. General Properties of Compact Sets. Sufficient Conditions for Compactness. Sequential Compactness. Compactness: Characterizations. Connected Spaces. Connectedness. Components. Totally disconnected spaces. Continuity. Continuity of Real valued functions. Continuous functions in arbitrary metric spaces. Equivalent Definitions of continuity and other characterizations. Results on continuity. Uniform Continuity. Continuous functions on compact spaces. Continuous functions on connected spaces. Path Connectedness. Equicontinuity and Arzela-Ascoli's Theorem. Open and Closed Maps. Homeomorphism. Banach Fixed Point Theorem and its Applications. Banach Contraction Theorem. Applications of Banach Contraction Principle. Bibliography.
Dhananjay Gopal has a doctorate in Mathematics from Guru Ghasidas University, Bilaspur, India and is currently Assistant Professor of Applied Mathematics in S V National Institute of Technology, Surat, Gujarat, India. He is author and co-author of several papers in journals, proceedings and a monograph on Background and Recent Developments of Metric Fixed Point Theory. He is devoted to general research on the theory of Nonlinear Analysis and Fuzzy Metric Fixed Point Theory.
Aniruddha Deshmukh is currently a student of (Integrated) MSc Mathematics and is associated to Applied Mathematics and Humanities Department, S V National Institute of Technology, Surat, Gujarat, India. He has been an active student in the department and has initiated many activities for the benefit of the students, especially as a member of the Science community (student chapter), known by the name of SCOSH. During his course, he has also attended various internships and workshop such as the Mathematics
Training and Talent Search (MTTS) Programme for two consecutive years (2017-2018) and has also done a project on the qualitative questions on Di erential Equations at Indian Institute of Technology (IIT), Gandhinagar, Gujarat, India in 2019. He has also qualified CSIR-NET JRF. Furthermore, his research interest focuses on Linear Algebra, Analysis, Topology and Geometry and their applicability in solving some real-world problems.
Abhay S Ranadive is a Professor at Department of Pure & Applied Mathematics, Ghasidas Vishwavidyalaya (A Central University), Bilaspur, Chattisgarh, India. He has been teaching at the University for last 30 years. He is author and co-author of several papers in journals and proceedings. He is devoted to general research on the theory of fuzzy sets and fuzzy logic, Modules and metric fixed point.
Shubham Yadav is currently a student of (Integrated) M.Sc. Mathematics and is associated to Applied Mathematics and Humanities Department, S V National Institute of Technology, Surat, Gujarat, India. As a member of SCOSH the student prominent science community in the institute, he has attended and organized various workshops and seminars. He also attended Madhava Mathematics Camp 2017. He did an internship on the calculus of fuzzy numbers at NIT, Trichy and one on operator theory at IIT, Hyderabad. He has also qualified for JRF. His main research interests are functional analysis and fuzzy sets with a knack for learning abstract mathematical concepts.