Analysis

We expect that the student has approved an undergraduate sequence in Analysis, this is usually achieved by taking the undergraduate courses MATE 4051, Advanced Calculus I and MATE 4052, Advanced Calculus II or by taking an upper level undergraduate course in Analysis.
Topics:

• Real numbers. Order axioms, axiom of completness.
• Basic topology in R: Compact sets, Bolzano-Weierstrass and Heine-Borel Theorems, connected sets.
• Sequences and series: monotone sequences, convergence tests, limit sup and limit inf, Cauchy sequences.
• Continuity: Basic definition, uniform continuity, types of discontinuities, limit of sequence of functions.
• Differentiation: Mean Value Theorem, Taylor Theorem, L'Hôpital.
• Riemann-Stieltjes integral: definition, basic properties, Fundamental Theorem of Calculus, Mean Value Theorem integral form. Functions of bounded variation.
• Sequences and series of functions. Absolute and uniform convergence, integration and differentiation, power series, Taylor series, radius of convergence.
• Functions of several variables. Differentiation, Inverse Function Theorem, Implicit function Theorem, Rank theorem.
• Integration in the Lebesgue sense. Measurable sets and measurable functions, Fatou’s Lemma, Monotone convergence Theorem and Dominated convergence Theorem.

Bibliography:

• Mathematical Analysis, T. M. Apostol, Second Ed., Addison-Wesley, 1974.
• Measure Theory, D.L. Cohn, Birkhäuser, 1994.
• The Elements of Real Analysis, R. G. Bartle, Second Ed., Wiley & Sons, 1976.
• Real Analysis: Modern Techniques and Their Applications, G.B. Folland, Wiley & Sons, 1999.
• Advanced Calculus, W. Fulks, Third Ed., Wiley & Sons, 1978.
• Elementary Analysis: The Theory of Calculus, Ross, A. Kenneth, Springer-Verlag, 1980.
• Principles of Mathematical Analysis, W. Rudin, McGraw-Hill, Third Ed., 1976.
• Introduction to Real Analysis, R. G. Bartle and D. R. Sherbert, Wiley & Sons, Third Ed., 2000.
Previous exams: