MATH 6360 - Fall 2018
Applied Analysis

    Course Info. View syllabus.
      Week 1. Review of metric spaces, completeness, characterization of compactness, extreme value theorem. Contraction mappings and fixed points.
        Homework Set 1 (Solutions), due Friday, August 31.
          Week 2. Applications of contractions mappings: integral equations, solutions to initial value problems. Local existence and uniqueness of solutions, stability.
            Homework Set 2 (Solutions), due Friday, September 7.
              Week 3. The contraction mapping theorem in multivariate calculus.
                Homework Set 3 (Solutions), due Friday, September 14.
                  Week 4. The inverse and implicit function theorems.
                    Homework Set 4 (Solutions), due Friday, September 21.
                      Week 5. Towards Lp spaces. Exam 1, Sep. 28. To see whether you are prepared, take a practice run (Solutions).
                        Week 6. Extending the Riemann integral to Lp spaces. Banach spaces.
                          Homework Set 5 (Solutions), due Friday, October 5.
                            Week 6. Dual spaces. Uniform boundedness
                              Homework Set 6 (Solutions), due Friday, October 19.
                                Week 6. Consequences of uniform boundedness for Fourier series and polynomial interpolation.
                                  Homework Set 7 (Solutions), due Friday, October 26.
                                    Weeks 7 and 8. Uniform convexity, best approximation propert and duality for Lp-spaces. Bounded inverse, closed graph theorem.
                                      Homework Set 8 (Solutions), due Friday, November 9.
                                        Week 9. Hilbert spaces. Exam 2, Nov 16. To see whether you are prepared, take a practice run (Solutions).
                                          Week 10. Homework Set 9 (Solutions), deferred to Monday, December 3.
                                            Week 11. Orthonormal bases and their characterization. Final exam, Dec 10. Review session: Wednesday, Dec 5, 9am, 646PGH. To see whether you are prepared, take a practice run (Solutions).

                                            MATH 6361 - Spring 2019
                                            Applied Analysis II

                                              Course Info. View syllabus.
                                                Week 1. Review of Hilbert spaces. Characterization of best approximation by orthogonal projection. Orthonormal bases.
                                                  Homework Set 1 (Solutions), due Friday, January 25.
                                                    Week 2. Fourier series. Convergence in L2 and pointwise convergence. Weak convergence.
                                                      Homework Set 2 (Solutions), due Friday, February 1.
                                                        Week 3. Relationships between weak and norm convergence. Weak compactness in Hilbert spaces.
                                                          Homework Set 3 (Solutions), due Friday, February 8.
                                                            Week 4. Linear and convex programming in Hilbert spaces. The Lax-Milgram theorem.
                                                              Homework Set 4 (Solutions), due Friday, February 22.
                                                                Week 5. The Hilbert-Schmidt norm and Hilbert-Schmidt operators. Exam 1, March 1. To see whether you are prepared, take a practice run (Solutions).
                                                                  Week 6. Compact self-adjoint operators. Towards the spectral theorem.
                                                                    Homework Set 5 (Solutions), due Friday, March 8.
                                                                      Week 7. The spectral theorem for compact, self-adjoint operators. Exam 1 retake, March 22. Material from Homework Sets 1-5.
                                                                        Homework Set 6 (Solutions), due Friday, March 29.
                                                                          Week 8. Diagonalizing normal operators. Fredholm theory.
                                                                            Homework Set 7 (Solutions), deferred to Monday, April 8.
                                                                              Week 9. Solutions to Schrödinger's eigenvalue problem and compact integral operators.
                                                                                Reading assignment. Review the Contraction Mapping Principle with the help of these notes and memorize the theorem until Monday.
                                                                                  Week 10. Calculus in normed vector spaces.
                                                                                    Homework Set 8, due Friday, April 19.
                                                                                      Reading assignment. Read the Inverse Function Theorem notes and memorize the theorem until Wednesday.
                                                                                        Week 11. The Inverse Function Theorem and applications. Introduction to the Calculus of Variations.
                                                                                          Homework Set 9, Part I, due Friday, April 26.
                                                                                            Exam 2, April 29. Material from Homework Sets 6-9.