Advanced Functional Analysis

Webpage of the Course MAT.405UB WS 2019/20

General Information

Welcome to the Practical Course of Advanced Functional Analysis MAT.405UB for the Master Degree in Mathematics at the University of Graz. This is the companion module to the theoretical course Advanced Functional Analysis MAT.404UB. Exercise sheets will be released every two weeks which will cover practicals aspect of the topics covered in MAT.404UB.

Please do not hesitate to email me with any questions you have regarding the module or the exercises.

Lectures Calendar

We have one lecture every two weeks. The lectures are two hours long, and are roughly every second Mondday from 10:00 to 12:00. In addition there will be a written exam at the end of the course. These are the dates:



Exercise Sheets

Due date Exercise Sheet Topics
14 October Sheet 1 \(L^p\) spaces. Lax-Milgram
28 October Sheet 2 Convolutions. Mollifiers. Sobolev spaces
11 November Sheet 3 Sobolev spaces. Elliptic PDEs. Partitions of unity
25 November Sheet 4 Sobolev embeddings. TVS. Minkowski functionals
9 December Sheet 5 Locally convex spaces. Weak topologies
20 January Sheet 6 Distributions. Compact operators. Spectral theory

Final Exam

The exam will contain 6 problems. You should choose 4 problems to solve, each graded from 0 to 25 points. You should state which problems you are going to solve. If more than 4 problems are solved, your final grade will be computed by summing the worst 4 scores. For grading, problems and sub-questions within the problems are considered independent. For example, if you only solve point (c) in Problem 1 and you have not solved (a) and (b), you will be awarded 5 points for Problem 1. The hints are there to help you, but of course you may solve the exercises in whichever way you prefer. You may use your solutions to the problems in the Exercise Course, as well as the notes from the course MAT.404UB. You will have 2 hours and 30 minutes to solve the exam.


Two weeks before Lecure x I will upload an exercise sheet on this page. This has to be solved by the day Lecture x happens. You are required to upload your solutions in Moodle as a single pdf file (scans of handwritten solutions or latex) before Lecture x starts.

There will be an exam at the end of the semester. The final percentage will be the average between exam points and exercise points (in percentage), and it will be translated into a grade according to the table below. A pass will be granted for a grade of 4 or better.

Percentage 0-49% 50-59% 60-74% 75-89% 90-100%
Grade 5 4 3 2 1

Note: Every student needs to present at least 3 times during the semester.