These notes originate from lectures given at the University of Maryland and at the Scuola Normale Superiore, Pisa, in the past two years.
The main purpose of the lectures was to give a first systematic, self contained account of the theory of Caratheodory's and Kobayashi's distances and differential metrics on domains in complex Banach spaces. A complete description of the behaviour of these metrics in concrete cases follows from 'Schwartz lemmas', and these depend on maximum principles. Thus, after two chapters devoted to the theory of holomorphic maps in Banach spaces, a third chapter is concerned with the investigation of various maximum principles and some of their consequences.