Functional Analysis
The aim is to equip students with tools in functional analysis to tackle advanced problems in mathematics and other fields.
Functional analysis is a fundamental branch of mathematics that extends the concepts of linear algebra and analysis to infinite-dimensional spaces. Its purpose is to develop tools and techniques to solve complex problems that arise in various areas of mathematics, physics, engineering, and beyond.
This course will cover the fundamental concepts, theorems, and techniques of functional analysis. Topics will include normed spaces, Banach spaces, Hilbert spaces, linear operators and functionals, the Hahn-Banach theorem, duality, compact and self-adjoint operators, and the spectral theorem. The course will emphasize rigorous mathematical reasoning and the development of problem-solving skills.
1. Review of metric and topological spaces
2. Normed spaces and Banach spaces
3. Linear operators
4. Uniform boundedness principle
5. Open mapping theorem
6. Closed graph theorem
7. Dual spaces
8. Hahn-Banach theorem
9. Adjoint and compact operators
10. Weak topology
11. Hilbert spaces
12. Riesz representation theorem
13. Orthonormal bases
14. Spectral decomposition
Exam: 50%, Homework: 50% (2 hours per week)
Single-variable and multi-variable calculus, linear algebra, B36 Real Analysis, A110 Measure Theory, or equivalent.
Walter Rudin, Functional Analysis, 2nd Edition, McGraw-Hill, 1991
Michael Reed and Barry Simon, Functional Analysis, Methods of Modern Mathematical Physics (Volume 1), Academic Press, revised edition, 1980.
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011th edition.
Kosaku Yosida, Functional Analysis, Springer, 6th edition, 1995.
Alternating Years Course, odd years