Since the early 70's,there has been intensive development in the theory of functions of an infinite number of complex variables. This has led to the establishment of completely new principles (e.g. concerning the behaviour of fixed points) and has thrown new light on some classical finite dimensional results such as the maximum principle, the Schwarz lemma and so on. Perhaps the most spectacular advances occurred in connection with the old problem of the determination of the holomorphic automorphisms of complex manifolds.

This book is based on the introductory lectures on this latter field delivered at the University of Santiago de Compostela in October 1981 by the authors. Originally, it was planned as a comprehensive postgraduate course relying on a deep knowledge of holomorphy in topological vector spaces and infinite dimensional Lie groups. However, seeing that some of the undergraduate students were mainly interested in the study of bounded domains in Banach spaces, the authors restricted their attention to these aspects. This proved to be a fortunate idea. We realized that by combining the methods of the theories developed independently by W. Kaup and J.P. Viguë with minor modifications, even the main theorems could be derived.

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