The aim of this course is to introduce postgraduate students to some of the fundamental notions and techniques of analysis, including advanced topics in functional analysis and and analysis on topological groups.

We will cover some of the following topics:

• Operators on topological vector spaces
• Weak topologies
• Convexity and fixed point theorems
• Spectral theorem for self-adjoint operators
• Commutative C*-algebras
• Topological groups
• Invariant measures on groups
• Representations of compact groups

Time: Friday 10-12pm; See TCC Timetable

Course Assessment: The course will be assessed by weekly homework sets posted here

Lecture plan:

• Lecture 1 (31/01): The Hahn-Banach theorem and applications.
• Lecture 2 (7/02): Completeness: uniform boundness principle and open map theorem.
• Lecture 3 (14/02): Uniform convergence and compactness theorem.

  A simple proof of Tichnoff theorem

• Lecture 4 (21/02): Riesz representation theorem.
• Lecture 5 (28/02): Convexity, Krein-Milman theorem, and applications.
• Lecture 6 (7/03): Haar measure.
• Lecture 7 (14/03): Spectral theory of selfadjoint operators,

  What does the spectral theorem says?

• Lecture 8 (21/03): Spectral theory II

Homework Problems:

• Problem Set 1 : submit solutions of 6 problems by February 14.
• Problem Set 2 : submit solutions of 4 problems by February 21.
• Problem Set 3 : submit solutions of 4 problems by February 28.
• Problem Set 4 : submit solutions of 4 problems by March 7.
• Problem Set 5 : submit solutions of 4 problems by March 14.
• Problem Set 6 : submit solutions of 4 problems by March 21.
• Problem Set 7 : submit solutions of 4 problems by March 28.

References:

• G. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
• D. Lax, Functional Analysis, Wiley lnterscience, New York, 2002.
• W. Rudin, Functional Analysis, McGraw-Hill, 1991.
• R. Zimmer, Essential Results in Functional Analysis, University of Chicago Press, 1990.