A Handbook of Real Variables, Softcover reprint of the original 1st ed. 2004 With Applications to Differential Equations and Fourier Analysis

The subject of real analysis dates to the mid-nineteenth century - the days of Riemann and Cauchy and Weierstrass. Real analysis grew up as a way to make the calculus rigorous. Today the two subjects are intertwined in most people's minds. Yet calculus is only the first step of a long journey, and real analysis is one of the first great triumphs along that road. In real analysis we learn the rigorous theories of sequences and series, and the profound new insights that these tools make possible. We learn of the completeness of the real number system, and how this property makes the real numbers the natural set of limit points for the rational numbers. We learn of compact sets and uniform convergence. The great classical examples, such as the Weierstrass nowhere-differentiable function and the Cantor set, are part of the bedrock of the subject. Of course complete and rigorous treatments of the derivative and the integral are essential parts of this process. The Weierstrass approximation theorem, the Riemann integral, the Cauchy property for sequences, and many other deep ideas round out the picture of a powerful set of tools.

Basics.- Sets.- Operations on Sets.- Functions.- Operations on Functions.- Number Systems.- Countable and Uncountable Sets.- Sequences.- to Sequences.- Limsup and Liminf.- Some Special Sequences.- Series.- to Series.- Elementary Convergence Tests.- Advanced Convergence Tests.- Some Particular Series.- Operations on Series.- The Topology of the Real Line.- Open and Closed Sets.- Other Distinguished Points.- Bounded Sets.- Compact Sets.- The Cantor Set.- Connected and Disconnected Sets.- Perfect Sets.- Limits and the Continuity of Functions.- Definitions and Basic Properties.- Continuous Functions.- Topological Properties and Continuity.- Classifying Discontinuities and Monotonicity.- The Derivative.- The Concept of Derivative.- The Mean Value Theorem and Applications.- Further Results on the Theory of Differentiation.- The Integral.- The Concept of Integral.- Properties of the Riemann Integral.- Further Results on the Riemann Integral.- Advanced Results on Integration Theory.- Sequences and Series of Functions.- Partial Sums and Pointwise Convergence.- More on Uniform Convergence.- Series of Functions.- The Weierstrass Approximation Theorem.- Some Special Functions.- Power Series.- More on Power Series: Convergence Issues.- The Exponential and Trigonometric Functions.- Logarithms and Powers of Real Numbers.- The Gamma Function and Stirling’s Formula.- An Introduction to Fourier Series.- Advanced Topics.- Metric Spaces.- Topology in a Metric Space.- The Baire Category Theorem.- The Ascoli-Arzela Theorem.- Differential Equations.- Picard’s Existence and Uniqueness Theorem.- The Method of Characteristics.- Power Series Methods.- Fourier Analytic Methods.- Glossary of Terms from Real Variable Theory.- List of Notation.- Guide to the Literature.

Completely self-contained, methodical exposition for the mathematically-inclined researcher; also valuable as a study guide for students Realistic, meaningful connections to ordinary differential equations, boundary value problems, and Fourier analysis Example-driven, incisive explanations of every important idea, with suitable cross-references for ease of use