MATH 4332/6313 - Spring 2018
Introduction to Real Analysis, II

    Course Info. View syllabus.
      Week 1. Metric spaces. Open and closed sets. Closure and interior.
        Homework Set 1 (Solutions), due February 1.
          Week 2. Compactness. Completeness.
            Homework Set 2 (Solutions), due February 8.
              Week 3. Compactness vs. sequential compactness.
                Homework Set 3 (Solutions), due February 15.
                  Week 4. Continuity and its characterization. Continuity and compactness.
                    Homework Set 4 (Solutions), due February 22.
                      Week 5. Metric completion. Baire's theorem.
                        Exam 1, March 1, in class. When you feel prepared, try a practice run (Solutions). Special office hours: Wed, Feb 28, 9:30-10:50am.
                          Week 6. The Contraction Mapping Theorem.
                            Homework Set 5 (Solutions), due March 8.
                              Week 7. Newtons's method and finding solutions to ordinary differential equations.
                                Homework Set 6 (Solutions), due March 22.
                                  Week 8. Polynomial approximation. Taylor polynomials and Taylor series.
                                    Homework Set 7 (Solutions), due March 29.
                                      Week 9. Weierstrass theorem.
                                        Exam 2, April 5, in class. When you feel prepared, try a practice run (Solutions).
                                          Week 10. Best approximation with polynomials. Equioscillation condition. Chebyshev polynomials.
                                            Homework Set 8 (Solutions), due April 19.
                                              Week 11. Approximation with trigonometric polynomials. Fourier series.
                                                Homework Set 9 (Solutions), due April 26.
                                                  Final Exam, May 8, 8am-11am. When you feel prepared, try a practice run (Solutions).

                                                  MATH 4331/6312 - Fall 2017
                                                  Introduction to Real Analysis

                                                    Course Info. View syllabus. Office hours: PGH 604, Tu 11:30am-12:30pm, We 1-2pm.
                                                      Week 1. The topology of Rn. Cauchy sequences and completeness. Open and closed sets.
                                                        Homework Set 1 (Solutions), due date deferred to September 7.
                                                          Week 2. Closure of a set. Compactness.
                                                            Homework Set 2 (Solutions), due September 14.
                                                              Week 3. Heine Borel property and other properties of compact sets. Limits and continuity of functions.
                                                                Homework Set 3 (Solutions), due September 21.
                                                                  Week 4. Discontinuous functions. Uniform continuity. Connected sets.
                                                                    Homework Set 4 (Solutions), due September 28.
                                                                      Week 5. Connectedness and the Intermediate Value Theorem in higher dimensions. Summary of the material in a handout.
                                                                        Exam 1, October 5, in class. When you feel prepared, try a practice run (Solutions).
                                                                          Week 6. Differentiablity and the Mean Value Theorem.
                                                                            Homework Set 5 (Solutions), due October 19.
                                                                              Week 7. The Riemann integral and its properties.
                                                                                Homework Set 6 (Solutions), due October 26.
                                                                                  Week 8. The Fundamental Theorem of Calculus, see handout. Normed vector spaces.
                                                                                    Homework Set 7 (Solutions), due November 2.
                                                                                      Week 9. Inner product spaces. Cauchy-Schwarz inequality. Relation between inner product and norm.
                                                                                        Exam 2, November 9, in class. Covers material from the Intermediate Value Theorem (and its higher-dimensional generalization) to normed vector spaces. When you feel prepared, try a practice run.
                                                                                          Week 10. Hölder and Minkowski inequalities for functions and sequences.
                                                                                            Homework Set 8 (Solutions), due Tuesday, November 21.
                                                                                              Week 11. Limits of sequences of functions. Uniform convergence. Completeness of C(K).
                                                                                                Homework Set 9 (Solutions), due November 30.
                                                                                                  Week 12. Equicontinuity. Total boundedness. Characterization of compact subsets in C(K).
                                                                                                    Final exam. Dec. 12, 11am-2pm, in classroom. Material from Homework Sets 1-9. When you feel prepared, try a practice run (Solutions). Review session on Tuesday, Dec 5, 10am-noon, PGH 646.