Wed 11/30: Spectral Representation for unbounded self adjoint operators.
See Reed and Simon, “Modern Methods in Mathematical Physics I: Functional Analysis”, Chapter 8.
Mon 11/28: Spectral Representation.
See Reed and Simon, “Modern Methods in Mathematical Physics I: Functional Analysis”, Chapter 7.
Wed 11/23: Applications of and commentary on the spectral theorem.
Mon 11/21: Spectral Theorem for bounded normal operators.
Fri 11/18: Stone-Weierstrass Theorem.
Wed 11/16: Gelfand transform for commutative Banach algebras.
c.f. Pedersen, “Analysis Now”, Ch. 4.2, for instance.
Mon 11/14: Banach space theorems: Uniform Boundedness and Closed Graph.
- Reading: Ch. 4.7, 4.12, 4.13.
Wed 11/9: Homework 4.
- Due Wednesday 11/16: Homework 4
Wed 11/9: Fredholm alternative, Baire Category and Open Mapping Theorem.
- Reading: Ch. 4.7, 4.12.
Mon 11/7: Index for Fredholm operators, consequences for spectral theory of compact operators.
Fri 11/4: Fredholm operators.
Wed 11/2: Spectral theory for compact normal operators, Fredholm operators.
Mon 10/31: Compact operators, spectral theory for compact normal operators.
Fri 10/28: Compact operators, spectral theory for compact normal operators.
- Reading: 8.2. At this point, I am mostly following the treatment in Pedersen’s “Analysis Now”, Chapter 3.3
Wed 10/26: Holomorphy of resolvent, compact operators.
- Reading: Chapter 7.5, 8.1, 8.2
Mon 10/24: Holomorphy of resolvent, weak topologies, Alaoglu’s Theorem.
- Reading: Chapter 7.5, 4.8, 4.9.
Fri 10/21: Homework 3.
- Due Friday 10/28: Problems (3.2) #8; (3.3) #7; (3.4) #6; (3.6) #4; (3.9) #9; (3.10) #4, 12, 15; (7.3) #2, 4, 6;
Fri 10/21: Spectral theory: properties of resolvent, spectrum.
- Reading: Chapter 7.3, 7.4, 7.6, 7.7.
Wed 10/19: Spectral theory: basic concepts.
- Reading: Chapter 7.1-7.3
- Here are some solutions to the midterm exam.
Fri 10/7: No Class.
Class will resume on Wednesday Oct 19. Have a good break!
Wed 10/5: Adjoints, Self-adjoint, unitary and normal operators.
- Reading: Chapter 3.10
- Midterm exam due
Mon 10/3: Riesz representation, Hilbert adjoint operator.
- Reading: Chapter 3.8, 3.9.
Fri 9/30: Orthonormal bases.
- Reading: Chapter 3.6
- Midterm Exam: Midterm take-home problems, due Wednesday 10/5.
Wed 9/28: Orthonormal sets.
- Reading: Chapter 3.4, 3.5.
Mon 9/26: Hilbert spaces.
- Reading: Chapter 3.3
Fri 9/23: Hilbert spaces.
- Reading: Chapters 3.1-3.2
Wed 9/21: Hahn-Banach Theorem and applications.
- Reading: Chapters 4.1-4.6
Mon 9/19: Lebesgue measure theory and L^p continued, Hahn-Banach Theorem.
- Reading: Chapters 4.1-4.3
Fri 9/16: Lebesgue measure theory and L^p continued.
Here are some notes (without proofs) on measure, integration and L^p.
Wed 9/14: Lebesgue measure theory and L^p.
Mon 9/12: Homework 2.
- Due Wednesday 9/21: Problems (2.4) #4; (2.6) #14, 15; (2.7) #6, 7, 8, 12, 13; (2.8) #6; (2.9) #13; (2.10) #6, 11.
Mon 9/12: Linear functionals and dual spaces continued.
- Reading: Chapters 2.8-2.10
Fri 9/9: Linear functionals and dual spaces.
- Reading: Chapters 2.8-2.10
Wed 9/7: Linear operators.
- Reading: Chapters 2.6-2.8
Fri 9/2: Classes cancelled.
See you on Wednesday 9/7.
Fri 9/2: Linear operators.
- Reading: Chapters 2.6-2.8
Wed 8/31: Finite dimensional normed spaces.
- Reading: Chapters 2.4, 2.5.
Mon 8/29: Vector spaces, Banach spaces.
- Reading: Chapters 2.1-2.3
Fri 8/26: Homework 1.
- Due Wednesday 9/7: Problems (1.1) #13, 15; (1.5) #11, 12; (1.6) #5; (2.1) #5; (2.2) #6, 10-12; (2.3) #3, 4, 9, 14.
Fri 8/26: Metric spaces, completeness completed?.
- Reading: Chapters 1.4-1.6.
Wed 8/24: Metric spaces, completeness, continued.
- Reading: Chapters 1.3, 1.4.
Mon 8/22: Metric spaces, completeness.
- Reading: Chapters 1.1-1.3.