Real Analysis and Foundations by Steven G. Krantz CRC Press, 1991, 0849371562, Hardcover without dust jacket, VG condition, no marks, no underlining, no highlighting, 295 pages. From the PREFACE Overview The subject of real analysis, or “advanced calculus,” has a central position in undergraduate mathematics education. Yet because of changes in the preparedness of students, and because of their early exposure to calculus (and therefore lack of exposure to certain other topics) in high school, this position has eroded. Students unfamiliar with the value of rigorous, axiomatic mathematics are ill-prepared for a traditional course in mathematical analysis. Thus there is a need for a book that simultaneously introduces students to rigor, to the need for rigor, and to the subject of mathematical analysis. The correct approach, in my view, is not to omit important classical topics like the Weierstrass Approximation Theorem and the Ascoli–Arzela Theorem, but rather to find the simplest and most direct path to each. While mathematics should be written “for the record” in a deductive fashion, proceeding from axioms to special cases, this is not how it is learned. Therefore (for example) I do treat metric spaces (a topic that has lately been abandoned by many of the current crop of analysis texts). I do so not at first but rather at the end of the book as a method for unifying what has gone before. And I do treat Riemann–Stieltjes integrals, but only after first doing Riemann integrals. I develop real analysis gradually, beginning with treating sentential logic, set theory, and constructing the integers. The approach taken here results, in a technical sense, in some repetition of ideas. But, again, this is how one learns. Every generation of students comes to the university, and to mathematics, with his or her own viewpoint and background. Thus I have found that the classic texts from which we learned mathematical analysis are often no longer suitable, or appear to be inaccessible, to the present crop of students. It is my hope that my text will be a natural source for modern students to learn mathematical analysis. Unlike other authors, I do not believe that the subject has changed; therefore I have not altered the fundamental content of the course. But the point of view of the audience has changed, and I have written my book accordingly. CONTENTS Preface 1 Logic and Set Theory 1.1 Introduction 1.2 “And” and “Or” 1.3 “Not” and “If-Then” 1.4 Contrapositive, Converse, and “Iff” 1.5 Quantifiers 1.6 Set Theory and Venn Diagrams 1.7 Relations and Functions 1.8 Countable and Uncountable Sets EXERCISES 2 Number Systems 2.1 The Natural Numbers 2.2 Equivalence Relations and Equivalence Classes 2.3 The Integers 2.4 The Rational Numbers 2.5 The Real Numbers 2.6 The Complex Numbers EXERCISES Appendix: Construction of the Real Numbers 3 Sequences 3.1 Convergence of Sequences 3.2 Subsequences 3.3 Lim sup and Lim inf 3.4 Some Special Sequences EXERCISES 4 Series of Numbers 4.1 Convergence of Series 4.2 Elementary Convergence Tests 4.3 Advanced Convergence Tests 4.4 Some Special Series 4.5 Operations on Series EXERCISES 5 Basic Topology 5.1 Open and Closed Sets 5.2 Further Properties of Open and Closed Sets 5.3 Compact Sets 5.4 The Cantor Set 5.5 Connected and Disconnected Sets 5.6 Perfect Sets EXERCISES 6 Limits and Continuity of Functions 6.1 Definition and Basic Properties of the Limit of a Function 6.2 Continuous Functions 6.3 Topological Properties and Continuity 6.4 Classifying Discontinuities and Monotonicity EXERCISES 7 Differentiation of Functions 7.1 The Concept of Derivative 7.2 The Mean Value Theorem and Applications 7.3 More on the Theory of Differentiation EXERCISES 8 The Integral 8.1 Partitions and the Concept of Integral 8.2 Properties of the Riemann Integral 8.3 Another Look at the Integral 8.4 Advanced Results on Integration Theory EXERCISES 9 Sequences and Series of Functions 9.1 Partial Sums and Pointwise Convergence 9.2 More on Uniform Convergence 9.3 Series of Functions 9.4 The Weierstrass Approximation Theorem EXERCISES 10 Special Functions 10.1 Power Series 10.2 More on Power Series: Convergence Issues 10.3 The Exponential and Trigonometric Functions 10.4 Logarithms and Powers of Real Numbers 10.5 The Gamma Function and Stirling’s Formula 10.6 An Introduction to Fourier Series EXERCISES 11 Functions of Several Variables 11.1 Review of Linear Algebra 11.2 A New Look at the Basic Concepts of Analysis 11.3 Properties of the Derivative 11.4 The Inverse and Implicit Function Theorems EXERCISES 12 Advanced Topics 12.1 Metric Spaces 12.2 Topology in a Metric Space 12.3 The Baire Category Theorem 12.4 The Ascoli-Arzela Theorem EXERCISES Bibliography Index nthdegree books