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Elements of Metric Spaces

Elements of Metric Spaces


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About the Book

This book is intended for students of mathematics at undergraduate level and beyond, besides reviving old concepts with detailed examples, topics like linear metric spaces, FK-spaces, paranormed spaces and measure of noncompactness are special features of this book. Some important old concepts overlooked by many authors, e.g. semimetric spaces, relative compactness, equicontinuity, lower (upper) semicontinuous functions, uniform boundedness principle and Arzela-Ascoli's theorem have also been incorporated to make it self-contained. Undergraduate and postgraduate students will find this textbook of utmost use.

Table of Contents:
Preliminaries. otations; Sets; Union, Intersection and De'Morgan's Law; Cartesian Product; Relations; Functions; Bounded and Continuous Functions; Equivalent Sets; Countable Sets; Partial Ordering; Well Ordering and Well Ordering Principle; Axiom of Choice; Completeness Axiom; Zorn's Lemma; Sequences and Series; Monotonic Increasing and Decreasing Sequences; Bounded, Convergent and Cauchy Sequences; Convergent and Absolutely Convergent Series; Sequences of Functions: Point Wise Convergence and Uniform Convergence; Convergence Almost Every Where; Essential Supremum; Lebesgue Outer Measure; Lebesgue Measure; Measurable Functions; Characteristic and Simple Functions; Lebesgue Integral; Lp(E)-Spaces; Inequalities. Matric and Semimetric Spaces+ Metric Spaces; Examples; Semimetric Spaces; Some Metric and Topological Concepts; Subspaces; Miscellaneous Examples; Exercises. Completeness+ Convergent and Cauchy Sequences; Complete Metric Spaces; Some Properties of Completeness; Dense and Nowhere Dense Sets; Categories and Baire's Category Theorem; Completion of Metric Spaces; Equivalence of Metrics; Topologic and Metrizable Spaces; Miscellaneous Examples; Exercises. Continuity+ Limit of Function; Continuous Functions; Characterizations of Continuous Functions; Some Further Results; Uniform Continuity; Relation Between Continuity and Homeomorphism; Continuous Function on a Topological Space; Semi-Continuous Functions; Uniform Boundedness Principle; Exercises. Compactness+ Total Boundedness; Examples; Sequential Compactness; Bolzano-Weierstrass Property; Heine-Borel Compactness; Finite Intersection Property; Relation between Compactness and Continuity; Relation between Compactness and Uniform Continuity; Relation between Compactness and Uniform Convergence; Arzela-Ascoli's Theorem; Exercises. Connectedness+ Separated Sets; Disconnected and Connected Sets; Properties of Connected Sets; Totally Disconnected Sets; Components; Connected Subset of R; Relation Between Connectedness and Continuity; Miscellaneous Examples; Excercises. Some Further Topics on Metric Spaces+ Contraction; Banach Fixed Point Theorem; Applications; Measures of Noncompactness; Exercises. Linear Metric Spaces+ Linear Spaces; Linear Transformations; Isomorphism; Linear Dependence and Linear Hull; Hamel Basis and Dimensionality; Convexity and other Related Concepts; Seminorms; Linear Topological Spaces and Linear Metric Spaces; Paranormed Spaces; FK and other Spaces; Schauder Basis and AK-Property; Exercises.


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Product Details
  • ISBN-13: 9781844640416
  • Publisher: Paths Publishing Group
  • Publisher Imprint: Paths International Ltd
  • Height: 215 mm
  • Width: 140 mm
  • ISBN-10: 1844640418
  • Publisher Date: 01 May 2005
  • Binding: Hardback
  • Weight: 158 gr


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