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Full Description
A Passage to Modern Analysis is an extremely well-written and reader-friendly invitation to real analysis. An introductory text for students of mathematics and its applications at the advanced undergraduate and beginning graduate level, it strikes an especially good balance between depth of coverage and accessible exposition. The examples, problems, and exposition open up a student's intuition but still provide coverage of deep areas of real analysis. A yearlong course from this text provides a solid foundation for further study or application of real analysis at the graduate level.
A Passage to Modern Analysis is grounded solidly in the analysis of $\mathbf{R}$ and $\mathbf{R}^{n}$, but at appropriate points it introduces and discusses the more general settings of inner product spaces, normed spaces, and metric spaces. The last five chapters offer a bridge to fundamental topics in advanced areas such as ordinary differential equations, Fourier series and partial differential equations, Lebesgue measure and the Lebesgue integral, and Hilbert space. Thus, the book introduces interesting and useful developments beyond Euclidean space where the concepts of analysis play important roles, and it prepares readers for further study of those developments.
Contents
Sets and functions
The complete ordered field of real numbers
Basic theory of series
Basic topology, limits, and continuity
The derivative
The Riemann integral
Sequences and series of functions
The metric space ${\bf R}^n$
Metric spaces and completeness
Differentiation in ${\bf R}^n$
The inverse and implicit function theorems
The Riemann integral in Euclidean space
Transformation of integrals
Ordinary differential equations
The Dirichlet problem and Fourier series
Measure theory and Lebesgue measure
The Lebesgue integral
Inner product spaces and Fourier series
The Schroeder-Bernstein theorem
Symbols and notations
Bibliography
Index.
