Functional analysis (spring 2025)
Lecturer: Eduard Ortega
Lectures
- Tuesdays 10:15-12:00 in Realfagbygget R51
- Thursdays 12:15-14:00 in Realfagbygget R51
- First Lecture Tuesday 7th January from 10:15-12:00 in Realfagbygget R51
Office hours
- By appointment.
Reading materials
A. Bowers and N.J. Kalton An Introductory Course in Functional Analysis
Also see Harald Hanche-Olsen's notes and, E. Lieb and M. Loss. Analysis. Graduate Studies, AMS, for a more elementary approach to L^p-spaces, duality of L^p-spaces and related material on Lebesgues spaces that we are going to discuss during the first weeks.
Lectures log
| Week | Description | Comments | Recomended exercises |
|---|---|---|---|
| week 2 | Chapter 1 in Bowers & Kalton | Review of basic notions on normed spaces. Topology on metric spaces. Baire category theorem | |
| week 3 | Chapter 4 Bowers & Kalton | Uniform Boundedness Principle and Banach-Steinhaus theorem. Fourier series. | 4.2, 4.6 and 4.9 |
| week 4 | Chapter 4 Bowers & Kalton | The open mapping theorem and Closed graph theorem | 4.19, 4.20 and 4.21 |
| week 5 | Chapter 2 Bowers & Kalton | Dual spaces of \(\ell_p\) spaces and function spaces of measure spaces \(L_p(\Omega,\mu)\) | 2.1 and 2.2 |
| week 6 | Chapter 3 Bowers & Kalton | Sublinear functionals. Hahn-Banach extensions theorems. | |
| week 7 | Chapter 3 Bowers & Kalton | Invariant Hahn-Banach theorem. Haar measure of compact abelian groups | 3.1, 3.9, 3.10 and 3.11 |
| week 8 | Chapter 3 Bowers & Kalton | Reflexive spaces. Adjoint of operators | |
| week 9 | Chapter 3 Bowers & Kalton | Direct sums of Banach spaces. Quotients of Banach spaces | |
| week 10 | Chapter 5 Bowers & Kalton | Topological spaces | |
| week 11 | Chapter 5 Bowers & Kalton | Topological vector spaces. Basis of balanced and absorbent open sets of 0. | |
| week 12 | Chapter 5 Bowers & Kalton | The geometric Hahn-Banach theorem | |
| week 13 | Chapter 5 Bowers & Kalton | The weak topologies. Weak convergent sequences. The weak *-topology. The Banach Alaoglu theorem. | |
| week 14 | Chapter 6 Bowers & Kalton | Compact operators on Banach spaces. Rank-Nullity theorem | |
| week 15 | Chapter 7 Bowers & Kalton | Hilbert spaces. Compact operators on Hilbert spaces |
Exam
The Exam will take place the 19th May. You are NOT allowed to take notes or books. The Exam will consist of 5 questions, each of them with the same value.
3 of this questions will be about the main theorems and concepts of the course.
2 questions will be a selection of the exercises in the book A. Bowers and N.J. Kalton An Introductory Course in Functional Analysis.
Selected exercises: 2.1, 2.14, 3.4, 3.5, 3.16, 4.1, 4.4, 4.6, 4.12, 4.21, 5.9, 5.20, 5.31, 5.35, 6.1, 6.11