The purpose of the course is to present some techniques and results of functional analysis with applications to nonlinear dispersive equations, particularly the nonlinear Schrödinger equation (NSLE). The course begins with the introduction of some analytical tools: Lebesgue spaces, Riesz-Thorin interpolation theorem, Fourier transform. In the second part of the course, those tools will be applied to the study of the linear Schrödinger equation (LSE), focusing on the analysis of dispersive properties of solutions to the associated Cauchy problem. Similar properties and analogous results will be discussed for another fundamental dispersive equation: the Korteweg-de Vries equation. The third part of the course is dedicated to the study of the NSLE, specifically addressing the well-posedness of the associated Cauchy problem in various functional spaces using fixed-point techniques and the dispersive properties analyzed earlier. The last part of the course is devoted to the study of singular solutions of the NSLE.

By the end of the course, the student will:
- have acquired and be able to demonstrate some fundamental results of modern analysis, such as the Riesz-Thorin Interpolation Theorem.
- be able to state and prove theorems on the existence, uniqueness, and stability of solutions to the Cauchy problem for the LSE.
be able to derive representation formulas and discuss the fundamental properties of solutions to the Cauchy problem for the LSE.
- state and prove theorems on the existence and uniqueness of solutions to the Cauchy problem for the NSLE.
- state and prove results on the stability of solutions to the Cauchy problem for the NSLE.