Mathematical Analysis

Mathematical Analysis: An Introduction is a textbook containing more than enough material for a year-long course in analysis at the advanced undergraduate or beginning graduate level. The book begins with a brief discussion of sets and mappings, describes the real number field, and proceeds to a treatment of real-valued functions of a real variable. Separate chapters are devoted to the ideas of convergent sequences and series, continuous functions, differentiation, and the Riemann integral. The middle chapters cover general topology and a miscellany of applications: the Weierstrass and Stone-Weierstrass approximation theorems, the existence of geodesics in compact metric spaces, elements of Fourier analysis, and the Weyl equidistribution theorem. Next comes a discussion of differentiation of vector-valued functions of several real variables, followed by a brief treatment of measure and integration (in a general setting, but with emphasis on Lebesgue theory in Euclidean space). The final part of the book deals with manifolds, differential forms, and Stokes' theorem, which is applied to prove Brouwer's fixed point theorem and to derive the basic properties of harmonic functions, such as the Dirichlet principle.

Contents

• Preface
• 1 Real Numbers 1
• 2 Sequences and Series 28
• 3 Continuous Functions on Intervals 55
• 4 Differentiation 74
• 5 The Riemann Integral 98
• 6 Topology 123
• 7 Function Spaces 155
• 8 Differentiable Maps 175
• 9 Measures 201
• 10 Integration 223
• 11 Manifolds 253
• 12 Multilinear Algebra 269
• 13 Differential Forms 285
• 14 Integration on Manifolds 297
• References 323
• Index 325