Mathematical Analysis, Volume 1

This text is an outgrowth of lectures given at the University of Windsor, Canada. One of our main objectives isupdatingthe undergraduate analysis as a rigorous postcalculus course. While such excellent books as Dieudonne's Foundations of Modern Analysisare addressed mainly to graduate students, we try to simplify the modern Bourbaki approach to make it accessible to suciently advanced undergraduates. (See, for example,x4of Chapter 5.) On the other hand, we endeavor not to lose contact with classical texts, still widely in use. Thus, unlike Dieudonn e, we retain the classical notion of a derivative as anumber(or vector), not a linear transformation. Linear maps are reserved for later (Volume II) to give a modern version ofdierentials. Nor do we downgrade the classical mean-value theorems (see Chapter 5,x2) or Riemann{Stieltjes integration, but we treat the latter rigorously in Volume II, inside Lebesgue theory. First, however, we present the modern Bourbaki theory ofantidierentiation(Chapter 5,x5.), adapted to an undergraduate course. Metric spaces (Chapter 3,x11.) are introduced cautiously, after then-spaceEn , with simple diagrams in E2 (rather thanE3), and many \advanced calculus"-type exercises, along with only a few topological ideas. With some adjustments, the instructor may even limit all to En o rE2 (but not just to the real line,E1), postponing metric theory to Volume II.We do not hesitate to deviate from tradition if this simplies cumbersome formulations, upalatable to undergraduates. Thus we found useful someconsistent, though not very usual, conventions(see Chapter 5,x1and the end of Chapter 4,x4), and anearly use of quantiers(Chapter 1, x1{3), even in formulating theorems. Contrary to some existing prejudices, quantiers are easily grasped by students after some exercise, and help clarify all essentials.