Math 421/510 - Functional Analysis - Spring 2018

Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Tuesday and Thursday 11:00 AM to 12:30 PM in Room 203 of Mathematics Building.
Office hours: Tuesday 10-11, Thursday 2-3 or by appointment.

Marker/TA : Tongou (Thomas) Yang
E-mail : toyang at math dot ubc dot ca

Weekly quizzes

• Quiz 1 held at the end of class on Thursday, January 11. Solution
• Quiz 2 held at the end of class on Thursday, January 18. Solution
• Quiz 3 held at the end of class on Thursday, January 25. Solution
• Quiz 4 held at the end of class on Thursday, February 8. Solution
• Quiz 5 held at the end of class on Thursday, February 15. Solution
• Quiz 6 held at the end of class on Thursday, March 8. Solution
• Quiz 7 held at the end of class on Thursday, March 15. Solution
• Quiz 8 held at the end of class on Thursday, March 22. Solution

Homework

• Weeks 1, 2 and 3
• Practice problems: Chapter 5, Section 5.1, Exercises 1-16 of the textbook. These are not to be turned in.
• Homework set 1: Exercises 6, 8, 9, 13. (due on Tuesday Jan 23 at the beginning of lecture) Homework 1 Solution
• Weeks 4 and 5
• Practice problems: Chapter 5, Section 5.2, Exercises 17-26 of the textbook. These are not to be turned in.
• Homework set 2 (due on Tuesday Feb 6 at the beginning of lecture)
• Homework 2 Solution
• Weeks 6 and 7
• Practice problems: Chapter 5, Section 5.5, Exercises 54-67 of the textbook. These are not to be turned in.
• Homework set 3: Exercises 55, 56, 57, 58, 62 (due on Tuesday Feb 27 at the beginning of lecture).
• Homework 3 Solution
• Weeks 9 and 10
• Practice problems: Chapter 5, Section 5.3, Exercises 27-42 of the textbook. These are not to be turned in.
• Homework set 4: Exercises 29, 30, 32, 33, 37, 38, 40. (due on Tuesday Mar 13 at the beginning of lecture)
• Homework 4 Solution
• Weeks 11 and 12
• Practice problems: Chapter 5, Section 5.4, Exercises 43-53 of the textbook. These are not to be turned in.
• Homework set 5 (due on Tuesday April 3 at the beginning of lecture)
• Homework 5 Solution
• Week 13

Week-by-week course outline

This section contains a summary of the material covered in class, arranged by week. The treatment of these topics in lecture may vary somewhat from that of the text. Please stay tuned for possible changes.

• Week 1:
• Linear spaces
• Examples of infinite dimensional linear spaces
• Week 2:
• Normed linear spaces
• Topology induced by a norm
• Finite-dimensional normed spaces
• Week 3:
• Separable spaces: examples and non-examples
• Non-compactness of the unit ball in infinite-dimensional normed spaces
• Hamel and Schauder bases
• Week 4:
• Hahn-Banach theorem: the real case
• An application of Hahn-Banach: the hyperplane separation theorem
• Week 5:
• The Minkowski functional
• Hahn-Banach theorem: the complex version
• Week 6:
• Hilbert spaces
• Examples of Hilbert spaces
• Closest point to a convex set in a Hilbert space
• Week 7:
• Orthogonal projections and orthogonal complements
• Orthonormal bases
• Bounded linear functionals on Hilbert spaces
• Riesz-Frechet representation theorem on Hilbert spaces
• Week 9:
• Applications of Hilbert space methods
• An application to PDE: Dirichlet problem
• Tensor product of Hilbert spaces
• Hilbert spaces in quantum mechanics
• Fock spaces, Boson and Fermion Fock subspaces
• Week 10:
• Linear operators on normed vector spaces
• Open mapping theorem
• Inverse mapping theorem
• Closed graph theorem
• Week 11:
• Uniform boundedness principle
• Applications to Fourier series
• Dual of Lp
• Space of continuous functions vanishing at infinity
• Riesz representation theorem
• Week 12:
• Locally convex spaces
• Topology generated by seminorms
• Weak and Weak-star topologies
• Banach-Alaoglu thoerem
• Week 13:
• Weakly convergent sequences
• Weak closure versus weak sequential closure
• Closed convex sets in weak and strong topologies
• Application of weak convergence: approximation of the Dirac delta