# Functional Analysis I Autumn 2018

Lecturer
Manfred Einsiedler
Coordinator
Andreas Wieser

The first lecture takes place on Wednesday (19.09.2018).

TimePlace
Monday 13-15HG G 3
Wednesday 8-10HG G 5

Here is a list of the topics we treated for each week. If not mentioned otherwise, an item was treated with proof. We refer to our main source (see references).

 Week 1 Introduction (a subset of Chapter 1) and definition of norms, the induced topology as well as some examples (Section 2.1.1 up to Lemma 2.5). Week 2 Monday: Lemma 2.5, Proposition 2.6, Section 2.1.2 and 2.1.3 where Theorem 2.20 was only proven in the sub-additive case. Wednesday: Sections 2.2.1 and 2.2.2 where only (2),(3),(7) in Example 2.24 were treated ((7) without proof). Proposition 2.35 stated and its proof started. Week 3 Monday: Proposition 2.35, Section 2.3.1, Theorem 2.40 (without proof), Lemma 2.46. Wednesday: Proposition 2.51 (without proof), Section 2.4 up to the statement of Proposition 2.59. Week 4 Monday: Proposition 2.59, Corollary 2.60, Lemma 2.63 in a special case, Sections 2.4.2 and 2.5.1. Wednesday: Section 3.1.1, Section 3.1.2 up to and including Theorem 3.13 (the proof is not finished yet). Week 5 Monday: The rest of Section 3.1.2 (excluding Corollary 3.26) and Section 3.1.3 (the main idea). Wednesday: Corollary 3.26 and Section 3.2 in the separable case (i.e.~without Theorem 3.44) excluding Corollary 3.42. Definition 3.45, the theorem on Haar measures and some examples. Week 6 Monday: Theorem 3.47, Theorem 3.54 and Theorem 3.57. The independent proof of Theorem 3.54 was skipped and the proof of Theorem 3.57 was started. Wednesday: Theorem 3.57 and then Section 3.5.1 up to roughly the middle of the proof of Lemma 3.74. Week 7 Monday: The rest of Section 3.5.1 (except for Lemma 3.75) then Section 3.5.2, Proposition 3.83 and Section 3.5.5 (up to roughly the middle of the proof of Theorem 3.80). Wednesday: The rest of the proof of Theorem 3.80. Theorem 4.12 and then the proof of Theorem 4.1 based on Theorem 4.12 (as in Exercise 4.16). The contents of 4.1.1 were sketched. Week 8 Monday: Sections 4.2.2, 4.2.3 (except for Corollary 4.30), definition of reflexivity. Wednesday: Sections 7.1.1, 7.1.2 Week 9 Monday: Section 7.1.3 (without the shorter proof of Theorem 2.32), Section 7.2.1. The definition of amenable groups (in the sense of Lemma 7.18) with first examples. Wednesday: Proposition 7.34, Proposition 7.36 (the construction of the non-negative part of g was done in all detail, the rest was sketched). Week 10 Monday: Riesz representation theorem (Section 7.4): proof of Theorem 7.44 in the compact case assuming Lemma 7.50. Wednesday: Chapter 8 up to (and including) Section 8.1.1. Week 11 Monday: Proposition 8.11, Proposition 8.27, Section 8.3. Wednesday: Section 8.4. Week 12 Monday: Some of the contents in Section 8.5 (several spaces of distributions and derivatives of distributions), Section 8.6 up to and including Theorem 8.73. Wednesday: Corollary 8.74 and Section 8.6.1. Week 13 Monday: A review of unitary representations. Definition of the Sobolev spaces (Definition 5.1). Wednesday: Section 5.1 and the very beginning of Chapter 6. Week 14 Monday: Everything up to and including Lemma 6.7, Proposition 6.11 and Lemma 6.10 (the latter without proof). Wednesday: Definition of the adjoint operator and self-adjoint operators. Theorem 6.27 (including the lemmas needed in the proof).

Active participation in the exercises classes can yield a bonus of up to 5% on the grade of the exam. The bonus is obtained by presenting exercises in the exercises classes where one presentation gives one point. For the maximal bonus at least 8 exercises need to be presented.

The exact procedure is the following:

• The exercise sheet number n is published on this webpage on the Thursday of the n'th week of the semester.
• You then have until Friday in week n+1, 12:00 to solve the exercise sheet and volunteer to present one or more exercises on the vorxn-page. Note that you also need to hand in the exercises for which you were selected by Friday in week n+1, 13:00. Each assistant has a box in HG J 68 for that purpose. Make sure that you keep a copy of the solutions you need to present for yourself so that you can prepare your talk based on them.
• Per exercise class and exercise a few students are selected for a presentation. Which exercise you are supposed to present can be seen in the vorxn shortly after the deadline at 12 o'clock.
• The exercises are then presented and discussed in the exercise class on the Monday of week n+2.

Sheet 0 none Solutions for sheet 0 This sheet repeats a number of topics from previous lectures and will be discussed in the exercise classes on the 24th of September. The vorxn-system will not be used for this sheet and the class on the 24th.
Sheet 1 28.09.2018, 12:00 Solutions for sheet 1 Normed vector spaces.
Sheet 2 05.10.2018, 12:00 Solutions for sheet 2 Quotient spaces and Banach spaces.
Sheet 3 12.10.2018, 12:00 Solutions for sheet 3 The theorems of Arzela-Ascoli and Stone-Weierstrass. Operators and their norms.
Sheet 4 19.10.2018, 12:00 Solutions for sheet 4 Inner product spaces, uniformly convex spaces, Banach algebras, Hardy space.
Sheet 5 26.10.2018, 12:00 Solutions for sheet 5 Frechet-Riesz representation theorem, Haar measure on the torus.
Sheet 6 02.11.2018, 12:00 Solutions for sheet 6 Characters on compact abelian groups and Fourier series.
Sheet 7 09.11.2018, 12:00 Solutions for sheet 7 Unitary representations of compact abelian groups; Banach-Steinhaus.
Sheet 8 16.11.2018, 12:00 Solutions for sheet 8 The Hahn-Banach Theorem; Baire category theorem and corollaries.
Sheet 9 23.11.2018, 12:00 Solutions for sheet 9 More on the Hahn-Banach theorem and its corollaries and on explicit dual spaces. Amenability.
Sheet 10 30.11.2018, 12:00 Solutions for sheet 10 Weak topologies. Riesz representation theorem.
Sheet 11 07.12.2018, 12:00 Solutions for sheet 11 Locally convex vector spaces.
Sheet 12 14.12.2018, 12:00 Solutions for sheet 12 Separation theorems in locally convex vector spaces and extreme points.
Sheet 13 none Solutions for sheet 13 Sobolev spaces as well as compact self-adjoint operators and their spectral theorem. This exercise sheet will not be discussed in the exercise classes. However, the topics it treats are an important part of the course and in particular relevant for the exam.

You can enrol for an exercise class here. Note that the first exercise classes will take place in the second week of the semester.

We encourage you to attend exercise classes and to hand in solutions to the exercises you solve. You may also hand in the solutions to the exercise which you were not selected for. Give them to your assistant in the exercise class or put them in your assistant's box in the room HG J 68 on Monday. Your solutions will corrected and returned in the exercise class you are enrolled in or, if not collected, returned to the box.

TimePlaceAssistant
Monday 09-10HG F 26.3Giuliano Basso
Monday 09-10HG E 21Emilio Corso
Monday 09-10HG F 26.5Maxim Gerspach
Monday 09-10HG G 26.5Subhajit Jana

As discussed in greater detail in class on the 10th of December, the exam will consist of two types of questions. 50-60% will be theory questions from our discussions in class (such as definitions, lemmas, theorems, proofs and interactions of these). The remaining 50-40% of the questions will be similar to problems from the weekly exercise sheets.

We will be using the book Functional Analysis, Spectral Theory, and Applications by Manfred Einsiedler and Thomas Ward. It is available via SpringerLink here.

Other useful, and recommended references include the following:

• Lecture Notes on "Funktionalanalysis I" by Michael Struwe
• Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
• Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011.
• Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.
• Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.