 # Functional Analysis (Summer 2021)

## General Information

Goal: We will study the abstract theory of functional spaces, with a focus on applications in spectral theory, partial differential equations, and quantum mechanics.

Audience : Bachelor students of Mathematics and Physics. Other participants will get "Schein" if pass the course.

Time and place: There will be live zoom lectures. The first lecture takes place on Tuesday April 13 at 12:15.
• Lectures: Tuesday 12:15-13:45 and Friday 10:15-11:45.
Zoom link (Meeting ID: 966 3174 0227, Passcode: 529938).
• Exercises: Friday 14:15-15:45.
• Tutorials: TBA .

References:

Exercises and Tutorials: There will be a homework sheet every week. Doing the homework is the best way to learn the course's materials and to prepare for the final exam.

The tutorial section is provided to help you in reviewing the lectures. You can bring up your questions, discuss complementary materials, and try some extra exercises.

• You can get up to 100 points in the final exam.
• You can get up to 10 points in the midterm exam.
• You can get 1 point for every homework sheet if you solve at least 50% problems in the sheet.
You need 50 points to pass the course and 85 points to get the final grade 1.0.

Exams: You can use your notes (lecture notes, homework sheets and tutorial materials).

## Preliminary contents of the course

• Topological spaces: Convergence and continuity. Vector spaces. Linear functionals and dual spaces. Hahn-Banach extension theorem.
• Metric spaces: Convergence and completeness. Baire Category Theorem.
• Banach spaces:Â Linear operators. Weak and weak* convergence. Uniform Boundedness Principle. Open Mapping Theorem. Closed Graph Theorem. Banachâ€“Alaoglu theorem. Uniform Convexity and Reflexive spaces.
• Lp-spaces: Measures and integration. Lp-spaces and basic properties.
• Hilbert spaces: Riesz representation theorem. Laxâ€“Milgram theorem. Parseval identity.
• Bounded operators on Hilbert space: The spectrum. Self-adjointness. Compact operators. Spectral theorem.

## Contents of the lectures

13.4.2021. Motivation. Chapter 1: Topological spaces. Open and closed sets. Convergence and continuity. Vector spaces.

16.4.2021. Any vector space has a basis. Zorn lemma. Topological vector spaces. Examples: normed spaces, test functions, distributions. Minkowski function of an open convex set. Hahnâ€“Banach theorem: analytic version. Recorded video

20.4.2021. Hahnâ€“Banach theorem: analytic version (continued). Hahnâ€“Banach theorem: geometric version. Recorded video

23.4.2021. Hahnâ€“Banach theorem: geometric version (continued). Chapter 2: Metric spaces. Examples. Topology in a metric space. Complete metric spaces. Baire Category Theorem. Recorded video

27.4.2021.Baire Category Theorem (continued). Compactness and sequential compactness. Borel-Lebesgue theorem. Recorded video

30.4.2021. Continuous functions and compact sets. Hahnâ€“Banach theorem: strict geometric version. Banach fixed point theorem. Completion of metric spaces. Recorded video

04.05.2021. Chapter 3: Banach spaces. Examples. Locally compact spaces are finite dimensional. Recorded video

07.05.2021. Separable spaces. Operators on Banach spaces. Completeness of L(X,Y). Uniform boundedness principle. Recorded video

11.05.2021. Consequences of Uniform boundedness principle. Hahn's proof of the Uniform boundedness principle. Open mapping theorem. Recorded video

14.05.2021. Product spaces. Closed graph theorem. Direct sum, projection and complemented subspaces. Quotient spaces. Recorded video

18.05.2021. Completeness of quotient spaces. Banach-Masuz theorem on quotients of l^1. Chapter 4: Dual spaces and weak topologies. Weak convergence. Recorded video

21.05.2021. Weak topology. Closed balls are closed in the weak topology. Open balls are not open in the weak topology in finite dimensional spaces. Recorded video

28.05.2021. Closed convex sets are weakly closed. Mazur lemma. Weak-* topology. Banach-Alaoglu theorem. Recorded video

01.06.2021. Banach-Alaoglu theorem (cont.). Reflexive spaces. Kakutani theorem. Recorded video

08.06.2021. Kakutani theorem (cont.) and Eberleinâ€“Smulian theorem. Uniform convex spaces. Recorded video

11.06.2021. Milman-Pettis theorem. Chapter 5: Lp spaces. Measures and integrations. Recorded video

15.06.2021. Lp are Banach spaces. Uniform convexity and reflexibility. Recorded video

18.06.2021. Uniform convexity and reflexibility (cont.). Riesz representation theorem for dual space (Lp)^* Recorded video

22.06.2021. Weak convergence in Lp. Convolution. Approximation of Dirac delta function. Recorded video

25.06.2021. C_c^infty is dense in Lp. Fundamental lemma of calculus of variations. Fourier transform. Plancherel theorem. Inverse Fourier transform. Recorded video

29.06.2021. Hausdorffâ€“Young inequality. Dual space of L^1. Recorded video

02.07.2021. Dual space and non-separability of L^infty. l^p spaces. Weak and strong convergences of l^1. Chapter 6: Hilbert spaces. Inner product. Parallelogram law. Recorded video

06.07.2021. Uniform convexity and reflexibility. Projection on closed convex subsets. Projection on closed subspaces. Riesz representation theorem. Recorded video

09.07.2021. Lax-Milgram theorem. Direct sum of orthogonal spaces. Bessel's inequality. Parseval's identity. Existence of orthonormal basis. Bounded operators. Spectrum. Recorded video

12.07.2021. Self-adjoint operators. Spectral properties of self-adjoint operators. Compact operators. Spectral theorem for compact self-adjoint operators. Recorded video

16.07.2021. Bra-ket notation. Spectral decomposition for compact operators. Examples of compact operators: finite-rank operators, Hilbert-Schmidt operators. Recorded video