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

1 | Aug 21—Aug 25 | Linear Spaces. Linear Maps. Algebra of linear maps. Index of a linear map. | Chapter 1, 2.1, 2.2 | Chapter 1: 1.7, 1.8, 2.1: 2.1.3, 2.2: 2.2.9 H1: H1 | Aug 30 |

2 | Aug 28—Sep 1 | Flag of nullspaces of powers of a linear operator. Nilpotent operators. Jordan canonical form. Index of a linear map. The Hahn-Banach Theorem. | 2.1, 2.2, 3.1 | Sep 6 | |

3 | Sep 4 | Labor Day - no class. | Sep 7 | ||

3 | Sep 6—Sep 8 | The Hahn-Banach Theorem. The extension theorem. Geometric Hahn-Banach theorem. Extensions of the Hahn-Banach theorem. Applications of the Hahn-Banach theorem. Extension of positive linear functions. Banach limits. Finitely additive invariant set functions. | 3.1, 3.2, 3.3, 4.1, 4.2, 4.3 | 3.2: 3.2.2, 3.3: 3.3.3 H2: H2 | Sep 13 |

4 | Sep 11—Sep 15 | Normed linear spaces. Norms. Noncompactness of the unit ball. Isometries. | 5.1, 5.2, 5.3 | Sep 20 | |

5 | Sep 18—Sep 22 | Hilbert space. Scalar product. Closest point in a closed convex subset. Linear functionals. Linear span. | 6.1, 6.2, 6.3, 6.4 | 5.1: 5.1.3 H3: H3 | Sep 27 |

6 | Sep 25—Sep 29 | Applications of Hilbert space results. Radon-Nikodym theorem. | 7.1 | H4: H4 | Oct 4 |

7 | Oct 2—Oct 6 | Measure theory review. Radon-Nikodym theorem. | 7.1 | Oct 11 | |

8 | Oct 9—Oct 13 | Dirichlet's problem. | 7.2, 7.3 | Oct 18 | |

9 | Oct 18 | Midterm 1. | |||

9 | Oct 16—Oct 20 | Dirichlet's problem. | 7.2, 7.3 | ||

10 | Oct 23—Oct 27 | Duals of normed linear spaces. Bounded linear functionals. Extension of bounded linear functionals. Reflexive spaces. Support function of a set. | 8.1, 8.2, 8.3, 8.4 | H5: H5 | Nov 1 |

11 | Oct 30—Nov 3 | Applications of duality. Completeness of weighted powers. The Müntz approximation theorem. | 9.1, 9.2 | Nov 9 | |

12 | Nov 6—Nov 10 | The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. | 10.1, 10.2, 10.3 | H6: H6 | |

12 | Nov 11 | Veteran's Day - no class. | |||

13 | Nov 13—Nov 17 | The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. | 10.1, 10.2, 10.3 | ||

14 | Nov 20—Nov 22 | The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. | 10.1, 10.2, 10.3 | ||

14 | Nov 23—Nov 26 | Thanksgiving recess. | |||

15 | Nov 27—Dec 1 | The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. | 10.1, 10.2, 10.3 | H7 (Midterm 2): H7 | |

15 | Nov 29 | Midterm 2. | |||

16 | Dec 4—Dec 6 | ||||

Finals Week | Dec 11 (Monday) | Final Exam, 1:00 pm - 3:00 pm (regular room). |