A Course in Mathematical Analysis, vol. 2: Metric and Topological Spaces, Functions of a Vector Variable

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A COURSE IN MATHEMATICAL ANALYSIS

A COURSE INMATHEMATICAL ANALYSIS, Volume II

Copyright

     D. J. H. Garling 2013

     ISBN 978-1-107-03203-3

     ISBN 978-1-107-67532-2

     QA611.28.G37 2013 514\'.325–dc23

     LCCN 2012044992

Contents

Introduction

Part III Metric and topological spaces

     11 Metric spaces and normed spaces

          11.1 Metric spaces: examples

          11.2 Normed spaces

          11.3 Inner-product spaces

          11.4 Euclidean and unitary spaces

          11.5 Isometries

          11.6 *The Mazur–Ulam theorem*

          11.7 The orthogonal group bold0mu mumu OdOdOdOdOdOd

     12 Convergence, continuity and topology

          12.1 Convergence of sequences in a metric space

          12.2 Convergence and continuity of mappings

          12.3 The topology of a metric space

          12.4 Topological properties of metric spaces

     13 Topological spaces

          13.1 Topological spaces

          13.2 The product topology

          13.3 Product metrics

          13.4 Separation properties

          13.5 Countability properties

          13.6 *Examples and counterexamples*

     14 Completeness

          14.1 Completeness

          14.2 Banach spaces

          14.3 Linear operators

          14.4 *Tietze\'s extension theorem*

          14.5 The completion of metric and normed spaces

          14.6 The contraction mapping theorem

          14.7 *Baire\'s category theorem*

     15 Compactness

          15.1 Compact topological spaces

          15.2 Sequentially compact topological spaces

          15.3 Totally bounded metric spaces

          15.4 Compact metric spaces

          15.5 Compact subsets of C(K)

          15.6 *The Hausdorff metric*

          15.7 Locally compact topological spaces

          15.8 Local uniform convergence

          15.9 Finite-dimensional normed spaces

     16 Connectedness

          16.1 Connectedness

          16.2 Paths and tracks

          16.3 Path-connectedness

          16.4 *Hilbert\'s path*

          16.5 *More space-filling paths*

          16.6 Rectifiable paths

Part IV Functions of a vector variable

     17 Differentiating functions of a vector variable

          17.1 Differentiating functions of a vector variable

          17.2 The mean-value inequality

          17.3 Partial and directional derivatives

          17.4 The inverse mapping theorem

          17.5 The implicit function theorem

          17.6 Higher derivatives

     18 Integrating functions of several variables

          18.1 Elementary vector-valued integrals

          18.2 Integrating functions of several variables

          18.3 Integrating vector-valued functions

          18.4 Repeated integration

          18.5 Jordan content

          18.6 Linear change of variables

          18.7 Integrating functions on Euclidean space

          18.8 Change of variables

          18.9 Differentiation under the integral sign

     19 Differential manifolds in Euclidean space

          19.1 Differential manifolds in Euclidean space

          19.2 Tangent vectors

          19.3 One-dimensional differential manifolds

          19.4 Lagrange multipliers

          19.5 Smooth partitions of unity

          19.6 Integration over hypersurfaces

          19.7 The divergence theorem

          19.8 Harmonic functions

          19.9 Curl

     B Linear algebra

          B.1 Finite-dimensional vector spaces

          B.2 Linear mappings and matrices

          B.3 Determinants

          B.4 Cramer\'s rule

          B.5 The trace

     C Exterior algebras and the cross product

          C.1 Exterior algebras

          C.2 The cross product

     D Tychonoff\'s theorem

     Index

     Contents for Volume I

     Contents for Volume III