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.

• Lecturer: Dr Silvio Fanzon
• Email: Silvio.Fanzon@uni-graz.at
• Office: Room 501, Department of Mathematics & Scientific Computing

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:

• Lecture 1: 14 October 2019
• Lecture 2: 28 October 2019
• Lecture 3: 11 November 2019
• Lecture 4: 25 November 2019
• Lecture 5: 9 December 2019
• Lecture 6: 20 January 2020
• Exam: 27 January 2020

Topics

• Linear operators: compact operators, spectral theory, unbounded operators and their adjoints, spectral theorem for self adjoint operators.
• Topology of vector spaces: locally convex spaces, weak topologies, compactness, reflexivity.
• Function spaces: \(L^p\) spaces, distributions, Sobolev spaces and embedding theorems.

References

• Functional Analysis: I will mainly follow Conway . Additional references will be taken from Rudin and Bressan
• Calculus in Normed Spaces: Mainly following Kesavan
• \(L^p\) spaces: I will follow the construction of Rudin
• Sobolev spaces: The main references are Leoni and Brezis
• Distributions: The excellent book by Mitrea

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.

• Exam Paper

Assessment

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.

• At the beginning of each class, a form will be handed out in which every student has to declare which problems they solved. Based on this, some students will be called at the blackboard to solve one of the exercises they declared. If the exercise is correctly solved, the student will get all the points corresponding to the declared exercises. The points awarded for each exercise will be specified in the exercise sheet.
• If any of the students is not able to present at the blackboard (for a valid reason), I will manually correct one exercise chosen at random among the declared ones; if it is properly solved, the student will get all the points from the declared exercises.
• If a student cannot come to the exercise lecture, I will correct some of the submitted exercises to grade him. Please, when doing so, write down in the submission box which exercises were solved.
• If a student declares an exercise that was not solved, there will be 0 points for that exercise sheet.

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.