• Unit Organiser: Alexander Gorodnik and Min Lee
• Class Time: Monday 11-12pm, Tuesday 12-1pm, Wednesday 11-12pm.
• Exercise Class: Wednesday 12-1pm every other week.
• Office hours: Monday 10-11am, Wednesday 10-11am and by appointment.

• Level: H/6 and M/7
• Credit point value: 20cp
• Year: 14/15
• Prerequisites: MATH 20200 Metric Spaces 2

Course Description: The unit aims to provide students with a firm grounding in the theory and techniques of Functional Analysis and to offer students ample opportunity to build on their problem-solving ability in this subject. It also aims to equip students with independent self-study and presentation-giving skills. This course sets out to explore some core notions in Functional Analysis which originated in the study of integral/differential equations and more generally equations for operators in infinite dimensional spaces. These techniques can be helpful, for instance, in analysing trigonometric series and can be used to make sense of the determinant of an infinite-dimensional matrix. Functional Analysis has found broad applicability in diverse areas of mathematics, physics, economics, and other sciences. Students will be introduced to the theory of Banach and Hilbert spaces. The highlight of the course will be an exposition of the four fundamental theorems in the Functional Analysis (Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, closed graph theorem). The unit may also include some discussion of the spectral theory of linear operators.

• Course Information

Lecture Notes:

• Lecture 1: Inner product spaces and normed spaces
• Lecture 2: Completeness - Banach and Hilbert spaces
• Lecture 3: Orthogonality and Fourier Series
• Lecture 4: Linear operators
• Lecture 5: Compact operators
• Lecture 6: Hahn-Banach theorem
• Lecture 7: Spectrum of linear operator

Homework Problems:

• Problem Set 1: due Feb. 4
• Problem Set 2: due Feb. 11
• Problem Set 3: due Feb. 18 ----- Solution to Problem 5
• Problem Set 4: due Feb. 25 ----- Solution to Problem 6
• Problem Set 5: due March 4 ----- Solution to Problem 6
• Problem Set 6: due March 11

References:

• E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1989.
• D. Lax, Functional Analysis, Wiley lnterscience, New York, 2002.
• W. Rudin, Functional Analysis, McGraw-Hill, 1991.
• N. Young, An Introduction to Hilbert Space, Cambridge University Press, 1988.
• R. Zimmer, Essential Results in Functional Analysis, University of Chicago Press, 1990.