Topics, Exams and Homework Assignments

Week Dates Topics Sections Covered Homework Due
1Aug 21—Aug 25 Linear Spaces. Linear Maps. Algebra of linear maps. Index of a linear map. Chapter 1, 2.1, 2.2Chapter 1: 1.7, 1.8, 2.1: 2.1.3, 2.2: 2.2.9
H1:  H1
Aug 30
2Aug 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, 3.1Sep 6
3Sep 4Labor Day - no class.Sep 7
3Sep 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, 4.33.2: 3.2.2, 3.3: 3.3.3
H2:  H2
Sep 13
4Sep 11—Sep 15 Normed linear spaces. Norms. Noncompactness of the unit ball. Isometries. 5.1, 5.2, 5.3Sep 20
5Sep 18—Sep 22 Hilbert space. Scalar product. Closest point in a closed convex subset. Linear functionals. Linear span. 6.1, 6.2, 6.3, 6.45.1: 5.1.3
H3:  H3
Sep 27
6Sep 25—Sep 29 Applications of Hilbert space results. Radon-Nikodym theorem. 7.1
H4:  H4
Oct 4
7Oct 2—Oct 6 Measure theory review. Radon-Nikodym theorem. 7.1Oct 11
8Oct 9—Oct 13 Dirichlet's problem. 7.2, 7.3Oct 18
9Oct 18Midterm 1.
9Oct 16—Oct 20 Dirichlet's problem. 7.2, 7.3
10Oct 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
11Oct 30—Nov 3 Applications of duality. Completeness of weighted powers. The Müntz approximation theorem. 9.1, 9.2Nov 9
12Nov 6—Nov 10 The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. 10.1, 10.2, 10.3
H6:  H6
12Nov 11 Veteran's Day - no class.
13Nov 13—Nov 17 The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. 10.1, 10.2, 10.3
14Nov 20—Nov 22 The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. 10.1, 10.2, 10.3
14Nov 23—Nov 26 Thanksgiving recess.
15Nov 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
15Nov 29Midterm 2.
16Dec 4—Dec 6
Finals WeekDec 11 (Monday)Final Exam, 1:00 pm - 3:00 pm (regular room).