Week | Dates | Topics | Sections Covered | Homework | Due |
---|---|---|---|---|---|

1 | Jan 10—Jan 12 | Review of weak- and weak-*-sequential convergence Riesz-Kakutani Representation Theorem (Chapter 8, Theorem 14). | Chapter 10, Chapter 8 | H8 : H8 | Jan 24 |

2 | Jan 17—Jan 19 | Applications of weak convergence. Approximations of the delta function by continuous functions. Divergence of Fourier series. Approximate quadrature. | 11.1, 11.2, 11.3 | ||

3 | Jan 22—Jan 26 | Weak and strong analyticity of vector-valued functions. Existence of solutions of partial differential equations. The representation of analytic functions with positive real part. The weak and weak* topologies. | 11.4, 11.5, 11.6, Chapter 12 | H9 : H9 | Jan 31 |

4 | Jan 29—Feb 2 | Locally convex topologies and the Krein-Milman Theorem. Separation of points by linear functionals. The Krein-Milman Theorem. | Chapter 13, 13.1, 13.2 | H10 : H10 | Feb 21 |

5 | Feb 5—Feb 9 | The Stone-Weierstrass Theorem. Choquet's Theorem. | Chapter 13, 13.3, 13.4 | ||

6 | Feb 12—Feb 16 | Positive functionals. Convex functions. Completely monotone functions. Theorems of Caratheodory and Bochner. | Chapter 14, 14.1, 14.2, 14.3, 14.4 | H11 : H11 | Feb 28 |

7 | Feb 19—Feb 23 | ||||

8 | Feb 26—Mar 2 | ||||

9 | Mar 5—Mar 9 | Spring Recess | NO CLASSES | ||

10 | Mar 12—Mar 16 | ||||

11 | Mar 19—Mar 23 | Banach algebras. Resolvent. Spectral radius. Functional calculus involving complex integration. Spectral mapping theorem. | Chapter 17 | ||

12 | Mar 26—Mar 30 | Spectral theory. Spectral measures. Projection values measures. | Chapter 31 | ||

13 | Apr 2—Apr 6 | Spectral theory. Spectral measures. Projection values measures. Classification of symmetric operators up to unitary equivalence. | Chapter 31 | H11 : H12 | Apr 18 |

17 | Apr 30—May 2 | GOAL: Chapter 30. |