The goal of this course is to introduce the foundations of differential calculus in Banach spaces, generalizing the most common theorems from finite-dimensional spaces (e.g., Taylor’s formula, the Implicit Function Theorem, and the Inverse Function Theorem). These tools will then be applied to the study of nonlinear problems.
In particular, we will discuss results in bifurcation theory (e.g., the Crandall–Rabinowitz Theorem) and degree theory (e.g., the Leray–Schauder mapping degree), as well as their direct applications to fluid equations (the water waves equation and the stationary Navier–Stokes equations).

Main bibliography:

• A. Ambrosetti, G. Prodi,  A Primer of Nonlinear Analysis;
• G. Teschl - Topics in Real and Functional Analysis;