The Hopf bifurcation theorem in Banach spaces

We prove a Hopf bifurcation theorem in general Banach spaces, which improves a classical result by Crandall and Rabinowitz. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular, our theorem can be applied to semilinear and quasi-linear partial differential equations in unbounded domains of $\mathbb{R}^n$.