Topics, Exams and Homework Assignments (528 B)

Week Dates Topics Sections Covered Homework Due
1Jan 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
2Jan 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
3Jan 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
4Jan 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
5Feb 5—Feb 9 The Stone-Weierstrass Theorem. Choquet's Theorem. Chapter 13, 13.3, 13.4
6Feb 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
7Feb 19—Feb 23
8Feb 26—Mar 2
9Mar 5—Mar 9 Spring Recess NO CLASSES
10Mar 12—Mar 16
11Mar 19—Mar 23 Banach algebras. Resolvent. Spectral radius. Functional calculus involving complex integration. Spectral mapping theorem. Chapter 17
12Mar 26—Mar 30 Spectral theory. Spectral measures. Projection values measures. Chapter 31
13Apr 2—Apr 6 Spectral theory. Spectral measures. Projection values measures. Classification of symmetric operators up to unitary equivalence. Chapter 31
H11 :  H12
Apr 18
14Apr 9—Apr 13 Spectral theory. Spectral measures. Projection values measures. Classification of symmetric operators up to unitary equivalence. Chapter 31
15Apr 16—Apr 20 Banach Algebras. Chapter 18, 19
16Apr 23—Apr 27 Banach Algebras. Chapter 18, 19
17Apr 30—May 2 Banach Algebras. Review. Chapter 18, 19
18May 7 An alternative Final Exam. Chapter 31, 18, 19.
H13 :  H13
May 7