A Course in Mathematical Analysis Volume II Metric and Topological Spaces Functions of a Vector Variable 1st Edition by Garling 1107032032 9781107032033
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Product details:
ISBN 10: 1107032032
ISBN 13: 9781107032033
Author: D.J.H. Garling
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 1 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 3 covers complex analysis and the theory of measure and integration.
Table of contents:
Part Three 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 Od
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 Four 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
Appendix 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
Appendix C Exterior algebras and the cross product
C.1 Exterior algebras
C.2 The cross product
Appendix D Tychonoff’s theorem
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