This text is the outgrowth of a course given at Berkeley since 1960.
The object is to redo calculus correctly in a setting of sufficient generality
to provide a reasonable foundation for advanced work in various branches
of analysis. The emphasis is on abstraction, concreteness, and simplicity.
A few abstract ideas are introduced, almost minimal in number. Such
important concepts as metric space, compactness, and uniform convergence
are discussed in such a manner that they will not need to be redone later.
They are given concrete illustration and their worth is demonstrated by
using them to prove the results of calculus, generalized in ways that are
obviously meaningful and practical.
The background recommended is any first course in calculus, through
partial differentiation and multiple integrals (although, as a matter of fact,
nothing is assumed except for the axioms of the real number system). A
person completing most of the material in this book should not only have
a respectable comprehension of basic real analysis but should also be ready
to take serious courses in such subjects as integration theory, complex
variable, differential equations, other topics in analysis, and general
topology. Experience indicates that this material is accessible to a wide
range of students, including many with primary interests outside mathe-
matics, provided there is a stress on the easier problems.
Author(s): Maxwell Rosenlicht
Publisher: Scott, Foresman
Year: 1968
Language: English
Pages: 266
CHAPTER I. NOTIONS FROM SET THEORY 1
1. Sets and elements. Subsets 2
2. Operations on sets 4
3. Functions 8
4. Finite and infinite sets 10
Problems 12
CHAPTER II. THE REAL NUMBER SYSTEM 15
1. The field properties 16
2. Order 19
3. The least upper bound property 23
4. The existence of square roots 28
Problems 29
CHAPTER III. METRIC SPACES 33
1. Definition of metric space. Examples 34
2. Open and closed sets 37
3. Convergent sequences 44
4. Completeness 51
5. Compactness 54
6. Connectedness 59
Problems 61
CHAPTER IV. CONTINUOUS FUNCTIONS 67
1. Definition of continuity. Examples 68
2. Continuity and limits 72
3. The continuity of rational operations. Functions with values in En 75
4. Continuous functions on a compact metric space 78
5. Continuous functions on a connected metric space 82
6. Sequences of functions 83
Problems 90
CHAPTER V. DIFFERENTIATION 97
1. The definition of derivative 98
2. Rules of differentiation 100
3. The mean value theorem 103
4. Taylor's theorem 106
Problems 108
CHAPTER VI. RIEMANN INTEGRATION III
1. Definitions and examples 112
2. Linearity and order properties of the integral 116
3. Existence of the integral 118
4. The fundamental theorem of calculus 123
5. The logarithmic and exponential functions 128
Problems 132
CHAPTER VII. INTERCHANGE OF LIMIT OPERATIONS 137
1. Integration and differentiation of sequences of functions 138
2. Infinite series 141
3. Power series 150
4. The trigonometric functions 156
5. Differentiation under the integral sign 159
Problems 160
CHAPTER VIII. THE METHOD OF SUCCESSIVE APPROXIMATIONS 169
1. The fixed point theorem 170
2. The simplest case of the implicit function theorem 173
3. Existence and uniqueness theorems for ordinary differential equations 177
Problems 190
CHAPTER IX. PARTIAL DIFFERENTIATION 193
1. Definitions and basic properties 194
2. Higher derivatives 201
3. The implicit function theorem 205
Problems 212
CHAPTER X. MULTIPLE INTEGRALS 215
1. Riemann integration on a closed interval in En. Examples and basic properties 216
2. Existence of the integral. Integration on arbitrary subsets of En. Volume 222
3. Iterated integrals 231
4. Change of variable 235
Problems 244
SUGGESTIONS FOR FURTHER READING 249
INDEX 251
