Introduction to Real Analysis MATH 5200-5210

1 Introduction to real Analysis math 5200-5210 Theodore KilgoreDate of most recent revision isJanuary 8, 2019 ContentsPrefacev1 Some Basic Sets .. upon sets .. concerning set operations .. Cartesian Product of two sets .. and Functions .. Logic .. , operations, and truth tables .. 1-142 Integers and Rational Introductory Remarks .. Basic properties of the integers .. properties of the integers .. properties of the integers .. Well Ordering Principle and Mathematical Induction and other representations of integers.

2 Properties of the integers .. A small excursion .. Construction of the rational numbers .. Inadequacy of the rational numbers .. 2-153 Building the real Introduction .. Sequences .. Sequences of rational numbers .. The real numbers .. Completeness .. 3-18iiiCONTENTS4 Finite series and sigma notation .. Tools the binomial theorem .. Infinite series .. Tools the geometric series .. A small excursion the definition ofe.. Some discussion of the exponential function.

3 More on convergence of series .. Root Test and the Ratio Test .. Convergence and the Alternating Series Test 4-145 Topological Basics .. A brief discussion of topology .. for a topology .. and Continuity .. Properties .. Topologies .. Connectedness .. Compactness .. Metric Spaces .. Some general results .. 5-106 Functions, limits, and Functions .. Limits of functions .. Continuity .. Uniform continuity .. lim sup and lim inf.

4 6-117 Finite and countable sets .. Uncountable sets .. 7-38 Representations of the real Introduction .. Decimal representation .. Binary representation .. Other representations .. The Cantor Set .. 8-49 Conclusion of math 5200, math 5210 begins9-1 CONTENTSiii10 The Derivative and the Riemann The Derivative .. Integrals .. The integral of a bounded non-negative function on abounded closed interval .. The integral of a bounded function on a bounded closedinterval.

5 The effect of unnatural ordering on integration .. The Riemann integral .. The linearity of the integral .. Shortcuts can go wrong, when defining the integral .. The Fundamental Theorem of Calculus .. Improper integrals and the integral test .. A problem with the Riemann integral .. 10-1611 Vector and Function Normed vector spaces .. Inner Products .. Norms for continuous functions .. Banach Spaces .. Linear Transformations and Continuity .. Weierstrass Approximation Theorem.

6 Linear operators defined by integral kernels .. Periodic functions and the Fourier series .. The F ej er operator .. Theorem for periodic functions .. 11-2312 Finite Taylor-Maclaurin Introductory remarks .. Finite Taylor expansions of a function .. 12-113 Functions given as Introductory remarks .. Functions defined as series .. Convergence of a power series .. Differentiation and integration of power series .. Double sums .. The rest of the story .. 13-914 Integrals on Integrals defined on Rectangles.

7 Fubini s Theorem .. 14-215 The Stieltjes A broader view of integration .. The Riemann-Stieltjes Integral .. 15-4ivCONTENTSP refaceThis document was begun as a text for math 5200, Spring 2012 and for MATH5210, Fall 2012. Since then, some of the topics in it have been expanded or havebeen subjected to stylistic revision. Continued attention has been devoted to theseemingly never-ending task of removing typographical errors. The documentis presented to the students and to their future instructors so that everyoneinterested in the matter can know what has been done in the course.

8 The currentversion is published at< kilgota>for in case that future revisions may be desirable, the students are encour-aged to keep the book in a binder. That ought to make it easy to add any newpages, and also make it easy to incorporate any portions which are revised orexpanded. Since the text is distributed in the form of a PDF file, students cankeep an electronic copy and print any part as needed. Or, if desired, a studentcan merely keep an electronic copy and bring it to class on a laptop, netbook, orother electronic device.

9 In any event, the pagination is done chapter-by-chapter,thereby making it possible to revise or expand any portion of the text withoutneed to revise the pagination of subsequent additional convenience, the text is distributed in two formats whichdiffer only in the dimensions of the pages. One of the two formats uses standardletter-sized pages, suitable for printing or for viewing on a desktop computerwith a large monitor. The second format uses short and wide pages, presentedin landscape mode. When thus arranged, the pages can be conveniently viewedwith no need for scrolling.

10 Thus, this second format is intended exclusively forviewing and not for perspectives for the studentsTypically, the students entering an introductory course in real Analysis havetaken the calculus sequence and one or both of an introductory differentialequations course and an introductory linear algebra course. To a too greatextent, these courses present a very rushed and too informal Introduction tohuge areas of mathematics. In these courses, the contents and the emphases arepredetermined by external constraints.