Chapman and Hall/CRC
200 pages | 50 B/W Illus.
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.
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.