Cover.pdf; Cover; A COURSE IN MATHEMATICAL ANALYSIS; Title; Copyright; 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 spaces13 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 spaces15.5 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 theorem17.5 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 manifolds19.4 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.