Let $(E, \| \cdot\|)$ be a normed vector space, and $f \colon \Omega \subseteq E \to E$ be a continuously differentiable map. We say that a fixed point $u$ of $f$ is hyperbolic if $Df(u) \colon E \to E$ is a hyperbolic linear dynamical system.

In this lecture we prove our first persistence result: if $u$ is a hyperbolic fixed point of $f$ then any nearby map $g$ has a unique fixed point near $u$. The proof is an application of the famous Banach Fixed Point Theorem.