Personal Homepage of Prof. Dr. Francesca Da Lio

Analysis IV ( Fourier Theory & Hilbert Spaces ), D-MATH

Spring Semester 2025

Lecturer: Prof. Francesca Da Lio

Exercise hours coordinator: Thomas Stucker

Diary of the lectures

#Week	Date	Content	Notes	Reference
1	20/21.02.2025	Slides of presentation of the course Motivation to the study of Hilbert Spaces. Definition of Inner Product Spaces. Examples. Norms. Cauchy-Schwarz inequality (proof). Parallelogram law (proof).Polarization identities (proof). Continuity of the inner product with respect to the product topology. Some topological definitions: open ball, interior point, open set, closed set, convex set, topological vector space,	Class Notes	Lecture Notes [Iac] until page 16.
2	27/28.02.2025	Comparison between norms on the same space. Differences between finite and infinite dimentional vector spaces. Examples of Hilbert Spaces. An example of a non separable Hilbert spaceOrthogonality. Gram-Schmid process. Algebraic basis. Orthonormal System. Bessel and Parseval Inequalities (Proof of Theorem 1.50). Definition of Hilbert Basis . Completeness Criterions (Proof of Theorem 1.52)	Class Notes	Lecture Notes [Iac] until page 25. For curiosity:. The cardinality of algebraic basis in complete normed Spaces
3	6/7.03.2025	Theorem of existence of an Hilbert basis. Every separable Hilbert space is isometric eithr to $C^n$ or $\ell_{C}. Theorem about the existence of the projection on closed subspaces and characterisation of the orthogonal projection. Statement and proof of the theorem about the projection on closed convex set. Characterization of the projection onto a convex set, with proof. Geometric intuition. The orthogonal space of a closed proper subspace of a Hilbert space.	Class Notes	Lecture Notes [Iac] until page 29.
4	13/14.03.2025	Nontriviality of the complement of a proper closed subspace. Projection onto a subspace in terms of a Hilbert basis of the subspace. Every closed subspace has an orthogonal complement. Proposition about isometry of H and Y⊕ Y^{\perp} for any closed subspace.Definition of linear and bounded operators, example of an unbounded operator (derivative). Definition of the norm of a linear operator and proof that it is equal to the Lipschitz constant. Continuity of linear operators if the dimension is finite. Example: the identity is not necessarily continuous if we change the topology. A linear operator between normed vector spaces is continuous if and only if it is bounded. Riesz' representation Theorem of continuous linear functionals on an Hilbert space, with proof. Corollary: canonical isometric isomorphism between a Hilbert space and its dual. Application: the Radon-Nikodym theorem.	Class Notes	Lecture Notes [Iac] until page 37.
5	20/21.03.2025	Von Neumann’s proof of the Radon-Nikodym theorem. Introduction to Fourier Series: Definition and Motovation. Proof of Theorem 2.4 (Fourier Basis) by applying Complex Stone-Weirstrass Theorem. Proof of Corollary 2.7. Exercise 2.2 and 2.3.	Class Notes	For the proof of Radon-Nikodym theorem see also the Class Notes. For a direct proof of Theorem 2.4 see e.g [SS]. Lecture Notes [Iac] until page 43.
6	27/28.03.2025	Basel's Problem. Fourier coefficients of a real-valued function (Proof of Proposition 2.12). Fourier coefficients of the derivative (Proof of Proposition 2.17). Asymptotic behavior of Fourier coefficients (Proof of Proposition 2.19). Proof of Corollaries 2.20 and 2.25. Summability implies regularity (Theorem 2.22, only statement). Proof of Theorem 2.26. Properties of Dirichlet Kernel.	Class Notes	Lecture Notes [Iac] until page 56
7	2/3.04.2025	Proof of Theorem 2.28, Proof of Riemann-Lebesgue Integral. An alternative statement of pointwise convergence under Dini conditions. Computation of Dirichlet Integral via Dirichlet Kernel and of the series \sum_k\frac{\sin(kx)}{k}. Derivation of Heat/Diffision Equation in 1-D. Method of separation of variables.	Class Notes	Lecture Notes [Iac] until page 64. For Computation of Dirichlet Integral see in polybox
8	9/10.04.2025	Proof of Theorem 2.34 (Existence of a solution to the Heat Equation under summability assumption of the FC of initial datum).Proof of Theorem 2.37 (Uniqueness of classical solution ). Non existence of solution for negatives times. Heuristic derivation of Fourier Transform Definition of FT for summable functions in R^d. Proof of Theorem 3.3.	Class Notes	Lecture Notes [Iac] until page 72

Recommended bibliography (Undergraduate-Master level):

[Iac] Lecture Notes, Mikaela Iacobelli.

[Bo] Méthodes mathématiques pour les sciences physiques, Jena-Michel Bony École polytechnique, 2000.

[Bre] Functional Analysis, Sobolev Spaces and Partial Differential Equations Haim Brezis, (Universitext) 2011th Edition.

[Ev] Partial Differential Equations" by Evans (American Mathematical Society, Laurence Craig Evans, AMS 2010 (2nd edition). [S] Partial Differential Equations in Action From Modelling to Theory, Sandro Salsa, Universitext (UTX), Springer. [SS] Real Analysis: Measure Theory, Integration, and Hilbert Spaces Elias M. Stein, Rami Shakarch, Princeton Lectures in Analysis Book 3.

[Y] An Introduction to Hilbert Spaces, Nicholas Young, Cambridge, Mathematical Textbooks, 1992.