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, due April 19.
                                              Week 11. Approximation with trigonometric polynomials. Fourier series.
                                                Homework Set 9, due April 26.

                                                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.