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.

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).

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

Further reading: