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摘要

Starting with the basic theorems of functional analysis, and diffe!:.,entiation. in normed spaces, the book goes on to analyticity and holomorphyin Part I I. Very clear accounts of the maximum modulus theoremin infinite dimensional holomorphy and of the relationship betweenFrtkhet holomorphy and Gateaux holomorphy are given. The radiusof boundedness and a brief introduction to subharmonic and plurisubharmonicfunctions close Part II. In Part II I various topologies onthe space H(U, F) of holomorphic functions' over an open subset of aninfinite dimensional normed siace are studied. Domains of holomorphy,.holomorphic convexity and the Cartan -Thullen theorem for infinitedimensional holomorphy are included, as well as the Levi problemin Banach spaces. Part III ends with a study of germs of holomorphicfunctions and a proof of the completeness of H(U,F) for theNachbin topology.