Aim of the course is to introduce the basic tools of linear and nonlinear functional analysis, and to show applications of such techniques in analysis. The course is divided into two parts: the first one concerns spectral theory of linear operators, whose goal is to extend the classical notion of spectrum of a matrix to an infinite dimensional setting. The second part of the course is an introduction to the methods of nonlinear analysis to find zeros of a nonlinear functional over an infinite dimensional space. In particular it gravitates around the implicit function theorem and its variants.

**Course contents:**

Part 1: Linear analysis

- Spectrum of a bounded linear operator

- Compact operators, their spectrum and Fredholm theory

- Functional calculus and spectral theorem of selfadjoint operators

- Unbounded operators

Part 2: Nonlinear analysis

- Differential Calculus in Banach Spaces

- Implicit Function Theorem and Continuity Method

- Lyapunov–Schmidt Reduction and Bifurcation

- Hard Implicit Function Theorem

- Degree theory

**References**

[1] Ambrosetti, Prodi: A primer of nonlinear analysis. Cambridge Studies in Advanced Mathematics, 34. Cambridge University Press, 1995

[2] Amrein: Hilbert space methods in quantum mechanics.Fundamental Sciences. EPFL Press, Lausanne; Boca Raton, FL, 2009.

[3] Brezis: Functional analysis, Sobolev spaces and partial differential equations.Universitext. Springer, New York, 2011.

[4] Chang: Methods in nonlinear analysis. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005

[5] Levy-Bruhl: Introduction à la théorie spectrale, Dunod, 2003.

[6] Reed, Simon: Methods of modern mathematical physics. I. Functional analysis. Academic Press, Inc., New York, 1980