An Introduction to Analysis
Description
An essential undergraduate textbook on algebra, topology, and calculus
An Introduction to Analysis is an essential primer on basic results in algebra, topology, and calculus for undergraduate students considering advanced degrees in mathematics. Ideal for use in a one-year course, this unique textbook also introduces students to rigorous proofs and formal mathematical writing--skills they need to excel.
With a range of problems throughout, An Introduction to Analysis treats n-dimensional calculus from the beginning-differentiation, the Riemann integral, series, and differential forms and Stokes's theorem-enabling students who are serious about mathematics to progress quickly to more challenging topics. The book discusses basic material on point set topology, such as normed and metric spaces, topological spaces, compact sets, and the Baire category theorem. It covers linear algebra as well, including vector spaces, linear mappings, Jordan normal form, bilinear mappings, and normal mappings.
Proven in the classroom, An Introduction to Analysis is the first textbook to bring these topics together in one easy-to-use and comprehensive volume.
- Provides a rigorous introduction to calculus in one and several variables
- Introduces students to basic topology
- Covers topics in linear algebra, including matrices, determinants, Jordan normal form, and bilinear and normal mappings
- Discusses differential forms and Stokes's theorem in n dimensions
- Also covers the Riemann integral, integrability, improper integrals, and series expansions
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Content
Preface ix
1 Algebraic Fundamentals 1
1.1 Sets and Numbers 1
1.2 Groups, Rings, and Fields 13
1.3 Vector Spaces 30
2 Topological Fundamentals 62
2.1 Normed Spaces 62
2.2 Metric Spaces 72
2.3 Topological Spaces 87
2.4 Compact Sets 98
3 Mappings 110
3.1 Continuous Mappings 110
3.2 Differentiable Mappings 129
3.3 AnalyticMappings 149
4 Linear Mappings 189
4.1 Endomorphisms 189
4.2 Inner Product Spaces 211
5 Geometry of Mappings 231
5.1 Inverse Mapping Theorem 231
5.2 Implicit Function Theorem 246
5.3 Rank Theorem 255
6 Integration 266
6.1 Riemann Integral 266
6.2 Calculation of Integrals 284
7 Differential Forms 304
7.1 Line Integrals 304
7.2 Differential Forms 318
7.3 Integrals of Differential Forms 338
Index 363
Index of Notation 369
