This innovative textbook bridges the gap between undergraduate analysis and graduate measure theory by guiding students from the classical foundations of analysis to more modern topics like metric spaces and Lebesgue integration. Designed for a two-semester introduction to real analysis, the text gives special attention to metric spaces and topology to familiarize students with the level of abstraction and mathematical rigor needed for graduate study in real analysis. Fitting in between analysis textbooks that are too formal or too casual, From Classical to Modern Analysis is a comprehensive, yet straightforward, resource for studying real analysis.
To build the foundational elements of real analysis, the first seven chapters cover number systems, convergence of sequences and series, as well as more advanced topics like superior and inferior limits, convergence of functions, and metric spaces. Chapters 8 through 12 explore topology in and continuity on metric spaces and introduce the Lebesgue integrals. The last chapters are largely independent and discuss various applications of the Lebesgue integral.
Instructors who want to demonstrate the uses of measure theory and explore its advanced applications with their undergraduate students will find this textbook an invaluable resource. Advanced single-variable calculus and a familiarity with reading and writing mathematical proofs are all readers will need to follow the text. Graduate students can also use this self-contained and comprehensive introduction to real analysis for self-study and review.
Author(s): Rinaldo B. Schinazi
Publisher: Springer
Year: 2018
Language: English
Pages: 261
Tags: Number Systems, Power Series, Metric Spaces, Measure Theory, Lebesgue Integral, Riemann Integral
Cover......Page 1
From Calculus to Analysis......Page 4
Copyright......Page 5
Preface......Page 8
Contents......Page 10
1.1 The Algebra of the Reals......Page 12
Exercises......Page 15
1.2 Natural Numbers and Integers......Page 17
Exercises......Page 22
1.3 Rational Numbers and Real Numbers......Page 24
Exercises......Page 32
1.4 Power Functions......Page 34
Exercises......Page 42
2.1 Sequences......Page 44
Exercises......Page 49
2.2 Monotone Sequences, Bolzano-Weierstrass Theorem, and Operations on Limits......Page 51
Exercises......Page 55
2.3 Series......Page 56
Exercises......Page 64
2.4 Absolute Convergence......Page 66
Exercises......Page 73
3.1 Power Series......Page 76
3.2 Trigonometric Functions......Page 78
Exercises......Page 88
3.3 Inverse Trigonometric Functions......Page 90
Exercises......Page 92
3.4 Exponential and Logarithmic Functions......Page 93
Exercises......Page 103
4.1 Power Series Expansions......Page 106
Exercises......Page 118
4.2 Wallis' Integrals, Euler's Formula, and Stirling's Formula......Page 119
Exercises......Page 129
4.3 Convergence of Infinite Products......Page 131
Exercises......Page 140
4.4 The Number π Is Irrational......Page 142
Exercises......Page 145
5.1 Continuity......Page 148
Exercises......Page 157
5.2 Limits of Functions and Derivatives......Page 158
Exercises......Page 166
5.3 Algebra of Derivatives and Mean Value Theorems......Page 168
Exercises......Page 176
5.4 Intervals, Continuity, and Inverse Functions......Page 178
Exercises......Page 185
6.1 Construction of the Integral......Page 188
Exercises......Page 199
6.2 Properties of the Integral......Page 200
Exercises......Page 210
6.3 Uniform Continuity......Page 212
Exercises......Page 216
7: Convergence of Functions......Page 218
Power Series......Page 224
Exercises......Page 228
Existence......Page 232
Uniqueness......Page 234
Exercises......Page 243
9: Countable and Uncountable Sets......Page 246
Exercises......Page 254
Further Reading......Page 256
List of Mathematicians Cited in the Text......Page 258
Index......Page 260
