Functional Analysis
The aim is to equip students with tools in functional analysis to tackle advanced problems in mathematics and other fields.
For students who are interested in deepening their understanding of analysis, with applications across various scientific and engineering disciplines.
• To demonstrate a solid understanding of the fundamental concepts of functional analysis, including normed spaces, Banach spaces, Hilbert spaces, linear operators.
• To describe the key theorems of functional analysis, such as the Hahn-Banach theorem, the uniform boundedness principle and the Riesz representation theorem.
• To apply the principles and techniques of functional analysis to solve problems in various contexts.
• To develop the ability to construct rigorous mathematical arguments and proofs within the framework of functional analysis.
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
Homework 90% and Presentation 10%
Single-variable and multi-variable calculus, linear algebra, B36 Real Analysis, A110 Measure Theory, or equivalent.
Kreyszig, "Introductory Functional Analysis with Applications”
Peter Lax, ”Functional Analysis”
Kosaku Yosida, ”Functional Analysis”
Haim Brezis, ”Functional Analysis”
Different faculty teach this course each year