Chapter Summaries
Introduction: A Glimpse at Set Theory 1
1. The Algebra of Sets, 1
Inclusion, intersection, union, complement, Cartesian product
2. Functions, 11
Transformations, composition of functions, inverse functions,
direct and inverse images
3. Finite and Infinite Sets, 23
Finite, countable, and infinite sets
1. The Real Numbers
4. Fields, 28
27
Definition, elementary property of fields
5. Ordered Fields, 34
Definition, properties of ordered fields, absolute value, intervals,
Archimedean ordered fields, nested intervals
6. The Real Number System, 45
Complete Archimedean fields, suprema and infkna, the Supremum
Principle, Dedekind cuts, the Ca&r set
II. The Topology of Cartesian Spaces 58
7. Cartesian Spaces, 59
Definition, algebra of vectors, inner product, norm, basic inequalities
8. Elementary Topological Concepts, 69
Open sets, closed sets, neighborhoods, intervals, The Nested
Intervals Theorem, Bolzano-Weierstrass Theorem, connected
sets
xi
xii CHAPTER SUMMARIES
9. The Theorems of Heine-Bore1 and Baire, 84
Compactness, Heine-Bore1 Theorem, Cantor Intersection
Theorem, Lebesgue Covering Theorem, Nearest Point Theorem,
Circumscribing Contour Theorem, Baire’s Theorem
10. The Complex Number System, 94
Definition and elementary properties
III. Convergence 98
11. Introduction to Sequences, 98
Definition, algebraic combinations, convergence
12. Criteria for the Convergence of Sequences, 111
Monotone Convergence Theorem, Bolzano-Weierstrass Theorem,
Cauchy Convergence Criterion
13. Sequences of Functions, 121
Convergence of a sequence of functions, uniform convergence,
norm of a function, Cauchy Criterion for Uniform Convergence
14. Some Extensions and Applications, 132
The limit superior, limit inferior, the Landau symbols 0, o,
Cesàro summation, double and iterated sequences
IV. Continuous Functions 146
15. Local Properties of Continuous Functions, 146
Definition of continuity, equivalent conditions, algebraic combinations
of functions, linear functions, continuity of linear
functions
16. Global Properties of Continuous Functions, 160
Global Continuity Theorem, preservation of connectedness,
Bolzano’s Intermediate Value Theorem, preservation of ~OILpactness,
maximum and minimum values are attained, continuity
of the inverse function, Uniform Continuity Theorem, Fixed
Point Theorem for Contractions
17. Sequences of Continuous Functions, 175
Interchange of limit and continuity, approximation theorems,
Bernstem polynomials, Weierstrass Approximation Theorem,
Stone Approximation Theorem, Stone-Weierstrass Theorem,
Tietze Extension Theorem, equicontinuity, Arzelà-Ascoli
Theorem
18. Limits of Functions, 195
Definitions of deleted and non-deleted limits, elementary prop
erties, limit superior, semi-continuous functions
V. Differentiation
CHAPTER SUMMARIES
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206
19. The Derivative in R, 206
Definition, Interior Maximum Theorem, Rolle’s Theorem,
Mean Value Theorem, Taylor% Theorem, applications, interchange
of limit and derivative
20. The Derivative in RP, 224
Directional derivative, partial derivatives, the derivative, the
Chain Rule, the Mean Value Theorem, interchange of the order
of differentiation, Taylor’s Theorem
21. Mapping Theorems and Extremum Problems, 249
Class C’, Approximation Lemma, Locally One-one Mapping
Theorem, Weak Inversion Theorem, Local Solvability Theorem,
Open Mapping Theorem, Inversion Theorem, Implicit Function
Theorem, extremum problems, location of extrema, second
derivative test, extremum problems with constraints, Lagrange%
Method
VI. Integration
22. Riemann-Stieltjes Integral, 275
Definition, Cauchy Criterion for Integrability, bilinearity of the
Riemann-Stieltjes integral, additivity over intervals, Integration
by Parts, integrability of continuous functions, sequences of
integrable functions, Bounded Convergence Theorem, Riesz
Representation Theorem
23. The Main Theorems of Integral Calculus, 300
First Mean Value Theorem, Differentiation Theorem, Fundamental
Theorem of Integral Calculus, Integration by Parts,
Second Mean Value Theorem, Change of Variable Theorem,
integrals depending on a parameter, differentiation under the
integral sign, Leibniz’s Formula, interchange of the order of
integration, integral form for the remainder in Taylor’s Theorem
24. Integration in Cartesian Spaces, 316
Content in a Cartesian space, definition of the integral, Cauchy
Criterion, properties of the integral, First and Second Integrability
Theorems, Mean Value Theorem for Integrals, reduction
to iterated integrals, the Jacobian Theorem, transformations
of integrals
25. Improper and Infinite Integrals, 341
Improper integral of unbounded functions, Cauchy principal
value, definition of infinite integrals, Cauchy Criterion, tests
for convergence, absolute convergence, uniform convergence,
infinite integrals depending on a parameter, infinite integrals of
sequences, Dominated Convergence Theorem, iterated infinite
integrals
274

22. Riemann-Stieltjes Integral, 275
Definition, Cauchy Criterion for Integrability, bilinearity of the
Riemann-Stieltjes integral, additivity over intervals, Integration
by Parts, integrability of continuous functions, sequences of
integrable functions, Bounded Convergence Theorem, Riesz
Representation Theorem
23. The Main Theorems of Integral Calculus, 300
First Mean Value Theorem, Differentiation Theorem, Fundamental
Theorem of Integral Calculus, Integration by Parts,
Second Mean Value Theorem, Change of Variable Theorem,
integrals depending on a parameter, differentiation under the
integral sign, Leibniz’s Formula, interchange of the order of
integration, integral form for the remainder in Taylor’s Theorem
24. Integration in Cartesian Spaces, 316
Content in a Cartesian space, definition of the integral, Cauchy
Criterion, properties of the integral, First and Second Integrability
Theorems, Mean Value Theorem for Integrals, reduction
to iterated integrals, the Jacobian Theorem, transformations
of integrals
25. Improper and Infinite Integrals, 341
Improper integral of unbounded functions, Cauchy principal
value, definition of infinite integrals, Cauchy Criterion, tests
for convergence, absolute convergence, uniform convergence,
infinite integrals depending on a parameter, infinite integrals of
sequences, Dominated Convergence Theorem, iterated infinite
integrals
274
Xiv CHAPTER SUMMARIES
VII. Infinite Series 375
26. Convergence of Infinite Series, 375
Definition, Cauchy Criterion, absolute convergence, rearrangements
of series, double series, Cauchy product of series
27. Tests for Convergence, 387
Comparison Tests, Root Test, Ratio Test, Raabe’s Test, Integral
Test, Abel’s Lemma, Dirichlet’s Test, Abel’s Test, Alternating
Series Test
28. Series of Functions, 405
Absolute and uniform convergence, continuity of the limit,
term-by-term integration of series, term-by-term differentiation
of series, tests for uniform convergence, Cauchy Criterion,
Weierstrass M-test, Dirichlet’s Test, Abel’s Test, power series,
radius of convergence, Cauchy-Hadamard Theorem, term-byterm
integration and differentiation of power series, Uniqueness
Theorem, multiplication of power series, Bernsteïn’s Theorem,
Abel% Theorem, Tauber’s Theorem
References, 422
Hints for Selected Exercises, 424
Index, 441
7he Elements of Real Analysis