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A Course in Mathematical Analysis

Book Description

The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and teachers. Volume I focuses on the analysis of real-valued functions of a real variable. This second volume goes on to consider metric and topological spaces. Topics such as completeness, compactness and connectedness are developed, with emphasis on their applications to analysis. This leads to the theory of functions of several variables. Differential manifolds in Euclidean space are introduced in a final chapter, which includes an account of Lagrange multipliers and a detailed proof of the divergence theorem. Volume III covers complex analysis and the theory of measure and integration.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Introduction
  6. Part Three Metric and topological spaces
    1. 11 Metric spaces and normed spaces
      1. 11.1 Metric spaces: examples
      2. 11.2 Normed spaces
      3. 11.3 Inner-product spaces
      4. 11.4 Euclidean and unitary spaces
      5. 11.5 Isometries
      6. 11.6 *The Mazur–Ulam theorem*
      7. 11.7 The orthogonal group Od
    2. 12 Convergence, continuity and topology
      1. 12.1 Convergence of sequences in a metric space
      2. 12.2 Convergence and continuity of mappings
      3. 12.3 The topology of a metric space
      4. 12.4 Topological properties of metric spaces
    3. 13 Topological spaces
      1. 13.1 Topological spaces
      2. 13.2 The product topology
      3. 13.3 Product metrics
      4. 13.4 Separation properties
      5. 13.5 Countability properties
      6. 13.6 *Examples and counterexamples*
    4. 14 Completeness
      1. 14.1 Completeness
      2. 14.2 Banach spaces
      3. 14.3 Linear operators
      4. 14.4 *Tietze’s extension theorem*
      5. 14.5 The completion of metric and normed spaces
      6. 14.6 The contraction mapping theorem
      7. 14.7 *Baire’s category theorem*
    5. 15 Compactness
      1. 15.1 Compact topological spaces
      2. 15.2 Sequentially compact topological spaces
      3. 15.3 Totally bounded metric spaces
      4. 15.4 Compact metric spaces
      5. 15.5 Compact subsets of C(K)
      6. 15.6 *The Hausdorff metric*
      7. 15.7 Locally compact topological spaces
      8. 15.8 Local uniform convergence
      9. 15.9 Finite-dimensional normed spaces
    6. 16 Connectedness
      1. 16.1 Connectedness
      2. 16.2 Paths and tracks
      3. 16.3 Path-connectedness
      4. 16.4 *Hilbert’s path*
      5. 16.5 *More space-filling paths*
      6. 16.6 Rectifiable paths
  7. Part Four Functions of a vector variable
    1. 17 Differentiating functions of a vector variable
      1. 17.1 Differentiating functions of a vector variable
      2. 17.2 The mean-value inequality
      3. 17.3 Partial and directional derivatives
      4. 17.4 The inverse mapping theorem
      5. 17.5 The implicit function theorem
      6. 17.6 Higher derivatives
    2. 18 Integrating functions of several variables
      1. 18.1 Elementary vector-valued integrals
      2. 18.2 Integrating functions of several variables
      3. 18.3 Integrating vector-valued functions
      4. 18.4 Repeated integration
      5. 18.5 Jordan content
      6. 18.6 Linear change of variables
      7. 18.7 Integrating functions on Euclidean space
      8. 18.8 Change of variables
      9. 18.9 Differentiation under the integral sign
    3. 19 Differential manifolds in Euclidean space
      1. 19.1 Differential manifolds in Euclidean space
      2. 19.2 Tangent vectors
      3. 19.3 One-dimensional differential manifolds
      4. 19.4 Lagrange multipliers
      5. 19.5 Smooth partitions of unity
      6. 19.6 Integration over hypersurfaces
      7. 19.7 The divergence theorem
      8. 19.8 Harmonic functions
      9. 19.9 Curl
    4. Appendix B Linear algebra
      1. B.1 Finite-dimensional vector spaces
      2. B.2 Linear mappings and matrices
      3. B.3 Determinants
      4. B.4 Cramer’s rule
      5. B.5 The trace
    5. Appendix C Exterior algebras and the cross product
      1. C.1 Exterior algebras
      2. C.2 The cross product
    6. Appendix D Tychonoff’s theorem
  8. Index
  9. Contents for Volume I
  10. Contents for Volume III