The first course in analysis which follows elementary calculus is a critical one for students who are seriously interested in mathematics. Traditional advanced calculus was precisely what its name indicates-a course with topics in calculus emphasizing problem solving rather than theory. As a result students were often given a misleading impression of what mathematics is all about; on the other hand the current approach, with its emphasis on theory, gives the student insight in the fundamentals of analysis. In A First Course in Real Analysis we present a theoretical basis of analysis which is suitable for students who have just completed a course in elementary calculus. Since the sixteen chapters contain more than enough analysis for a one year course, the instructor teaching a one or two quarter or a one semester junior level course should easily find those topics which he or she thinks students should have. The first Chapter, on the real number system, serves two purposes. Because most students entering this course have had no experience in devising proofs of theorems, it provides an opportunity to develop facility in theorem proving. Although the elementary processes of numbers are familiar to most students, greater understanding of these processes is acquired by those who work the problems in Chapter 1. As a second purpose, we provide, for those instructors who wish to give a comprehen sive course in analysis, a fairly complete treatment of the real number system including a section on mathematical induction.
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ISBN-13
978-1-4615-9990-6 (9781461599906)
DOI
10.1007/978-1-4615-9990-6
Schweitzer Klassifikation
1 The real number system.- 1.1 Axioms for a field.- 1.2 Natural numbers, sequences, and extensions.- 1.3 Inequalities.- 1.4 Mathematical induction-definition of natural number.- 2 Continuity and limits.- 2.1 Continuity.- 2.2 Theorems on limits.- 2.3 One-sided limits-continuity on sets.- 2.4 Limits at infinity-infinite limits.- 2.5 Limits of sequences.- 3 Basic properties of functions on ?1.- 3.1 The Intermediate-value theorem.- 3.2 Least upper bound; greatest lower bound.- 3.3 The Bolzano-Weierstrass theorem.- 3.4 The Boundedness and Extreme-value theorems.- 3.5 Uniform continuity.- 3.6 Cauchy sequences and the Cauchy criterion.- 3.7 The Heine-Borel and Lebesgue theorems.- 4 Elementary theory of differentiation.- 4.1 Differentiation of functions on ?1.- 4.2 Inverse functions.- 5 Elementary theory of integration.- 5.1 The Darboux integral for functions on ?1.- 5.2 The Riemann integral.- 5.3 The logarithm and exponential functions.- 5.4 Jordan content.- 6 Metric spaces and mappings.- 6.1 The Schwarz and Triangle inequalities-metric spaces.- 6.2 Fundamentals of point set topology.- 6.3 Denumerable sets-countable and uncountable sets.- 6.4 Compact sets and the Heine-Borel theorem.- 6.5 Functions defined on compact sets.- 6.6 Connected sets.- 6.7 Mappings from one metric space to another.- 7 Differentiation in ?N.- 7.1 Partial derivatives.- 7.2 Higher partial derivatives and Taylor's theorem.- 7.3 Differentiation in ?N.- 8 Integration in ?N.- 8.1 Volume in ?N.- 8.2 The Darboux integral in ?N.- 8.3 The Riemann integral in ?N.- 9 Infinite sequences and infinite series.- 9.1 Elementary theorems.- 9.2 Series of positive and negative terms-power series.- 9.3 Uniform convergence.- 9.4 Uniform convergence of series-power series.- 9.5 Unorderedsums.- 9.6 The Comparison test for unordered sums-uniform convergence.- 9.7 Multiple sequences and series.- 10 Fourier series.- 10.1 Formal expansions.- 10.2 Fourier sine and cosine series-change of interval.- 10.3 Convergence theorems.- 11 Functions defined by integrals.- 11.1 The derivative of a function defined by an integral.- 11.2 Improper integrals.- 11.3 Functions defined by improper integrals-the Gamma function.- 12 Functions of bounded variation and the Riemann-Stieltjes integral.- 12.1 Functions of bounded variation.- 12.2 The Riemann-Stieltjes integral.- 13 Contraction mappings and differential equations.- 13.1 Fixed point theorem.- 13.2 Application of the fixed point theorem to differential equations.- 14 Implicit function theorems and differentiable maps.- 14.1 The Implicit function theorem for a single equation.- 14.2 The Implicit function theorem for systems.- 14.3 Change of variables in a multiple integral.- 14.4 The Lagrange multiplier rule.- 15 Functions on metric spaces.- 15.1 Complete metric spaces.- 15.2 Convex sets and convex functions.- 15.3 Arzela's theorem: extension of continuous functions.- 15.4 Approximations and the Stone-Weierstrass theorem.- 16 Vector field theory. The theorems of Green and Stokes.- 16.1 Vector functions on ?1 arcs, and the moving trihedral.- 16.2 Vector functions and fields on ?N.- 16.3 Line integrals.- 16.4 Green's theorem.- 16.5 Surfaces in ?3-parametric representation.- 16.6 Area of a surface and surface integrals.- 16.7 Orientable surfaces.- 16.8 The Stokes theorem.- 16.9 The Divergence theorem.- Appendices.- Appendix 1: Absolute value.- Appendix 2: Solution of inequalities by factoring.- Appendix 3: Expansions of real numbers in an arbitrary base.
