  In this paper we begin the study of some important Banach spaces of slice
hyperholomorphic functions, namely the Bloch, Besov and weighted Bergman
spaces, and we also consider the Dirichlet space, which is a Hilbert space. The
importance of these spaces is well known, and thus their study in the framework
of slice hyperholomorphic functions is relevant, especially in view of the fact
that this class of functions has recently found several applications in
operator theory and in Schur analysis. We also discuss the property of
invariance of this function spaces with respect to Moebius maps by using a
suitable notion of composition.
