Aim of the course is to introduce the basic tools of linear and nonlinear functional analysis, and to apply these techniques to problems in PDEs. 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 introduces the methods of nonlinear analysis to find the zeros of a nonlinear functional on a Banach 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 applications

- Lyapunov–Schmidt Reduction and Bifurcation

- Degree theory

**References**

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

[2] Bogachev, Smolyanov: Real and Functional Analysis. Moscow Lectures, Springer 2020

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

[4] Eidelman, Milman, Tsolomitis. Functional analysis. Graduate studies in Mathematics, 66. American Mathematical Society, 2004

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

[6] Teschl: Topics in Linear and Nonlinear Functional Analysis, Graduate Studies in Mathematics, Volume XXX, Amer. Math. Soc., Providence, link