# Functional Analysis I Autumn 2017

Lecturer
Prof. Dr. Alessandro Carlotto
Course Assistant
Mario Schulz
Teaching Assistants
Yash Jhaveri
Kathrin Näf
Alessandro Pigati
Shengquan Xiang
Lectures
Mon 13–15 / HG G 3
Wed 08–10 / HG G 5
Exercise classes
Mon 09–10
Office hours
Fr 13–15 or 14–16
Thu 14–16 or 16–18
First Lecture
20.09.2017
First Exercise class
25.09.2017
Course Catalogue
401-3461-00L Functional Analysis I

### Primary references

Michael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14.

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

### Extra references

Michael Reed and Barry Simon. Methods of Modern Mathematical Physics – Volume 1 (Functional Analysis). Academic Press, 1981

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. 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.

### Useful resources

Paul R. Halmos. How to write Mathematics.

Susan Hermiller, Melanie Martin and Eric York. Learning Calculus.

DateContentReference 1 (Struwe)Reference 2 (Brezis)Extras
20.09.2017 General presentation of the course and its scopes.
Textbooks and other resources.
–  – What is
Functional Analysis?
25.09.2017 The Baire Lemma, contextualization and proof.
Two questions in Real Analysis.
§ 1.1,  § 1.2,  § 1.3
(up to Def. 1.3.1.)
§ 2.1  –
27.09.2017 Uniform boundedness principle in metric spaces.
Baire category, different notions of smallness.
Satz 1.4.3 and § 1.3 § 2.2  –
02.10.2017 Normed and Banach spaces, examples, equivalent and inequivalent norms.
Closed subspaces, non-compactness of the unit sphere in a Banach space.
§ 2.1  –  –
04.10.2017 Continuity criteria for linear maps.
The Banach algebra of continuous, linear maps.
§ 2.2
(up to Satz 2.2.5)
–  –
09.10.2017 Series of linear, bounded operators. Spectral radius. Invertibility.
Product of Banach spaces. Intersections of Banach subspaces.
§ 2.2,  § 2.5  –  –
11.10.2017 Quotient of a Banach space by a closed subspace.
Hilbert spaces, orthogonal complements, projections.
§ 2.3,  § 2.4 § 5.1  –
16.10.2017 Orthonormal systems, Bessel's inequality, abstract Fourier series.
Hilbertean bases, the separability criterion, isometric classification.
– § 5.4 Hilbertean bases
and applications
18.10.2017 The Banach-Steinhaus theorem, a counterexample with an incomplete domain.
The open mapping principle, linear homeomorphisms between Banach spaces.
§ 3.1,  § 3.2 § 2.2,  § 2.3  –
23.10.2017 The closed graph theorem. The Töplitz criterion for symmetric maps.
The case of unbounded operators, the continuous inverse theorem.
§ 3.3 § 2.3  –
25.10.2017 Closable operators. Bounded operators are closable.
Example of a non-closable operator. Smooth differential operators are closable.
§ 3.4  –  –
30.10.2017 The Hahn-Banach theorem over the real and the complex fields.
Extension of linear, continuous functionals.
§ 4.1 § 1.1  –
01.11.2017 The dual of a Banach space, basic facts.
Geometric versions of Hahn-Banach. Annihilator of a subspace.
§ 4.2 § 1.2,  § 2.5  –
06.11.2017 Duality in Hilbert spaces: the Riesz representation theorem.
Bilinear forms and the Lax-Milgram theorem.
§ 4.3 § 5.2,  § 5.3  –
08.11.2017 Duality for $$L^p$$ spaces when $$p$$ is finite, discussion about $$L^\infty$$.
Uniform convexity of the unit ball: the Clarkson inequalities.
§ 4.4 § 4.3  –
13.11.2017 The Minkowski functional of an open, convex set. Separation of convex sets.
Extremal points, an existence result and the Krein-Milan theorem.
§ 4.5 § 1.2  –
15.11.2017 Weak convergence, basic facts and examples.
The weak topology on a normed space, different notions of closure.
§ 4.6 § 3.1,  § 3.2  –
20.11.2017 The bidual of a normed space. The canonical embedding of a normed space into its bidual.
Reflexivity: definitions, examples, criteria. A closed subspace of a reflexive space is reflexive.
§ 5.1 § 3.5  –
22.11.2017 Separability: examples and basic criteria.
The weak*-topology on the dual of a normed space. The Banach-Alaoglu theorem.
§ 5.2,  § 5.3
(up to Satz 5.3.2)
§ 3.6,  § 3.4  –
27.11.2017 The Eberlein-Smulyan compactness theorem. Minimizing distance from a point to a convex set.
Lower semicontinuity, coercivity and the direct method of the Calculus of Variations, examples.
§ 5.3,  § 5.4 § 3.5,  § 3.6  –
29.11.2017 The dual (adjoint) of a linear operator. The adjunction is an isometry on bounded operators.
Orthogonality relations and solvability criteria for linear equations in Banach spaces.
§ 6.1,  § 6.2
(up to Def. 6.2.1)
§ 2.6,  § 2.7  –
04.12.2017 Compact operators, basic properties. Operators of the form $$\mathrm{id}-T$$ with $$T$$ compact.
Two strong compactness theorems: Arzelà-Ascoli and Fréchet-Kolmogorov.
§ 6.2,  § 6.3 § 6.1,  § 6.2  –
06.12.2017 Dual vs. adjoint operator in Hilbert spaces. Symmetric and self-adjoint operators, examples.
Resolvent and spectrum of a linear map. The spectrum is closed. Properties of the resolvent.
§ 6.4,  § 6.5
(up to Def. 6.5.3)
§ 6.3  –
11.12.2017 Partition of the spectrum: point, continuous, residual. Three examples. Characterization
of the spectral radius for bounded operators. Functional calculus for rational functions.
§ 6.5  –  –
13.12.2017 Eigenvalues of symmetric operators are real, but their spectrum may equal $$\mathbb{C}$$.
Self-adjoint operators have real spectrum and related results.
§ 6.6  –  –
18.12.2017 Normal operators, characterization of their spectral radius. The spectral theorem
for compact, self-adjoint operators. Courant-Fischer characterization of eigenvalues.
§ 6.7 § 6.3,  § 6.4  –
20.12.2017 Spectral theory over the real field: two perspectives and their equivalence.
The Dirichlet spectrum of the Laplacian: some results and open problems.
–  – Complex to
real spectrum
Date
25 January 2018
Time
14:00–17:00
Place
HG F 3 (initials A–L),   HG F 5 (initials M–Z).
Date
07 August 2018
Time
09:00–12:00
Place
HG D 1.2
 Rules of the exam: Rules.pdf Advice for the exam: Advice.pdf Probeprüfung: Probeprüfung and Solutions of the Probeprüfung Training problems: Training.pdf

### office hours

DayDateTimeLocationAssistant
Thursday11.01.201814:00–16:00 HG G 19.2 Mario Schulz
Thursday18.01.201814:00–16:00 HG G 19.2 Alessandro Pigati
Monday22.01.201814:00–16:00 HG G 19.2 Alessandro Pigati,
Mario Schulz

Please register in myStudies and use echo.ethz.ch to enroll for an exercise class.

AssistantClassroom
Yash Jhaveri HG G 26.1
Kathrin Näf HG F 26.3
Alessandro Pigati HG F 26.5
Shengquan Xiang HG E 21

Every Monday, a new problem set is uploaded here. You have one week time to solve the problems. The following Monday during exercise class the corresponding solutions are discussed and you are invited to hand in your work for grading. If you have difficulties understanding or solving certain tasks, please prepare your questions and join the office hours on Thursdays or Fridays.

Every problem is marked by one of the following symbols.

Computation
Get your hands dirty and calculate.
Bookkeeping
Apply what you learn in basic situations.
Comprehension
Construct examples and give full proofs.
Hard problem
Challenging problems are denoted by one up to three diamonds. It is recommended that you start working on these problems only after you have reviewed the weekly material and carefully solved all other exercises in the assignment.
20.09.2017 25.09.2017 Problem Set 1 Solution 1
25.09.2017 02.10.2017 Problem Set 2 Solution 2
02.10.2017 09.10.2017 Problem Set 3 Solution 3
09.10.2017 16.10.2017 Problem Set 4 Solution 4
16.10.2017 23.10.2017 Problem Set 5 Solution 5 5.1 (b): $$A$$ and $$B$$ must be symmetric.
23.10.2017 30.10.2017 Problem Set 6 Solution 6 The tools needed to solve problems 6.3 – 6.7 will be covered on Wednesday, 25.10.2017.
30.10.2017 06.11.2017 Problem Set 7 Hints 7 Solution 7
06.11.2017 13.11.2017 Problem Set 8 Hints 8 Solution 8 Uniform convexity
13.11.2017 20.11.2017 Problem Set 9 Hints 9 Solution 9
20.11.2017 27.11.2017 Problem Set 10 Hints 10 Solution 10
27.11.2017 04.12.2017 Problem Set 11 Hints 11 Solution 11
04.12.2017 11.12.2017 Problem Set 12 Hints 12 Solution 12
11.12.2017 18.12.2017 Problem Set 13 Hints 13 Solution 13

UPDATE: From November on, office hours are on Thursdays and Fridays.

You are free to come and ask questions. The location changes depending on which assistant is on duty. The schedule is as follows (up to possible short-term changes, please check for updates):

DayDateTimeLocationAssistant
Thursday21.09.201714:00–16:00 HG F 28.3 Mario Schulz
Tuesday26.09.201713:00–15:00 HG FO 28.7 Kathrin Näf
Thursday28.09.201716:00–18:00 HG FO 27.9 Alessandro Pigati
Tuesday03.10.201713:00–15:00 HG G 38.1 Shengquan Xiang
Thursday05.10.201714:00–16:00 HG G 28 Yash Jhaveri
Tuesday10.10.201713:00–15:00 HG FO 28.7 Kathrin Näf
Thursday12.10.201716:00–18:00 HG FO 27.9 Alessandro Pigati
Tuesday17.10.201713:00–15:00 HG G 38.1 Shengquan Xiang
Thursday19.10.201714:00–16:00 HG G 28 Yash Jhaveri
Tuesday24.10.201713:00–15:00 HG FO 28.7 Kathrin Näf
Thursday26.10.201716:00–18:00 HG FO 27.9 Alessandro Pigati
Thursday02.11.201714:00–16:00 HG G 28 Yash Jhaveri
Friday03.11.201714:00–16:00 HG G 38.1 Shengquan Xiang
Thursday09.11.201716:00–18:00 HG FO 27.9 Alessandro Pigati
Friday10.11.201713:00–15:00 HG F 28.3 Mario Schulz
Thursday16.11.201714:00–16:00 HG G 28 Yash Jhaveri
Friday17.11.201714:00–16:00 HG G 38.1 Shengquan Xiang
Thursday23.11.201716:00–18:00 HG FO 27.9 Alessandro Pigati
Friday24.11.201713:00–15:00 HG FO 28.7 Kathrin Näf
Thursday30.11.201714:00–16:00 HG G 28 Yash Jhaveri
Friday01.12.201714:00–16:00 HG G 38.1 Shengquan Xiang
Thursday07.12.201716:00–18:00 HG FO 27.9 Alessandro Pigati
Friday08.12.201713:00–15:00 HG FO 28.7 Kathrin Näf
Thursday14.12.201714:00–16:00 HG G 28 Yash Jhaveri
Friday15.11.201714:00–16:00 HG G 38.1 Shengquan Xiang
Thursday21.12.201714:00–16:00 (update) HG F 28.3 Mario Schulz
Friday22.12.201713:00–15:00 HG FO 28.7 Kathrin Näf