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Complex analysis on infinite dimensional spaces. (English) Zbl 1034.46504
Springer Monographs in Mathematics. London: Springer (ISBN 1-85233-158-5/hbk). xv, 543 p. (1999).
This book presents an exhaustive exposition of the main developments in infinite-dimensional holomorphy in the last two decades. As the author writes in the first lines of his preface, infinite-dimensional holomorphy is the study of holomorphic and analytic functions over complex topological vector spaces. The definitive step in the creation of complex infinite-dimensional analysis was taken by Volterra in 1887. Hadamard realized the importance of Volterra’s work, and he had a strong influence on Fréchet and Gateaux. H. von Koch introduced a monomial approach to holomorphy on infinite-dimensional polydiscs which was developed by Hilbert in 1909. The current definition of holomorphic mapping was given independently by Graves and A. E. Taylor in the late 30’s. Analyticity played a relevant role in operator and spectral theory. M. A. Zorn made a number of important contributions in the mid-40’s. The topological vector structure of the space of holomorphic functions defined on an open subset of $${\mathbb C}^n$$ was studied by Grothendieck and Köthe (1953) and Martineau (1966). Nachbin, in a series of articles between 1966 and 1970, introduced and studied several topologies on the space of holomorphic functions on an infinite-dimensional space which are analyzed in Dineen’s book.
The author had written a book on the same subject before [“Complex analysis in locally convex spaces” (North-Holland Math. Stud. 57, North-Holland, Amsterdam) (1981; Zbl 0484.46044)]. Like that volume, the present one also emphasizes properties of different topologies defined on spaces of holomorphic functions and studies the relations between them. However, in the last 20 years considerable progress has been made in the study of polynomials and tensor products. This development is clearly reflected in the book. Polynomials are mainly a tool, but the first two chapters are a self-contained study of them, their duality, geometry and topologies. Here tensor products play an important role. These two chapters reflect that polynomials are nowadays an independent area of research within (multi-)linear functional analysis.
The present book is independent of the previous one. The latter is not necessary to understand the new one, which is self-contained. The central theme of the book is the space $$H(U)$$ of holomorphic functions on an open subset $$U$$ of a complex locally convex space $$E$$ and the relationships among three topologies on this space: the compact open topology $$\tau_0$$, the ported topology $$\tau_{\omega}$$ of Nachbin, and the topology $$\tau_{\delta}$$ which is generated by countable open covers of $$U$$. Each of these topologies is in general finer than the preceding one in the order they were listed above. The topology $$\tau_{\delta}$$ is much more difficult to describe, but it has much better locally convex properties than the compact open topology. The difficulties in the treatment of different topologies in the case of holomorphic functions defined on (say) an infinite-dimensional Banach space $$E$$ arise for the following reason: the space $$H(U)$$ must simultaneously contain copies of a space of type $$H(G)$$ for an open subset $$G$$ in the complex plane, which is a nuclear space in the sense of Grothendieck, and the infinite-dimensional Banach space $$E'$$. This makes the structure of $$H(U)$$ very complicated. The ideal situation occurs when $$\tau_0 = \tau_{\delta}$$. Chapter 4 of the book includes important examples when this is the case. Dineen writes in the preface: “We may regard $$\tau_{\omega}$$ as a compromise between the conflicting suggestions of several complex variables theory and linear functional analysis.” Many positive results about the coincidence of two of these three topologies have been obtained recently (some of them by the author himself) and are presented in the book. “Good properties” which play a relevant role are isolated. These include approximation properties, certain properties of Banach spaces, the condition (DN) of Vogt, the quasinormability of Grothendieck, the property (BB) for pairs of Fréchet spaces of J. Taskinen [Ann. Acad. Sci. Fenn., Ser. A I, Math., Diss. No. 63 (1986; Zbl 0612.46069)], and the locally convex properties for operators of A. Peris [Ann. Acad. Sci. Fenn., Ser. A I, Math. 19, 167–203 (1994; Zbl 0789.46006)].
Complex analysis on infinite-dimensional spaces consists of 6 chapters. As we have explained before, the first two deal with polynomials and spaces of polynomials. The author has adopted an integrated approach to the development of polynomials using multilinear mappings and tensor products. Here the thesis of R. A. Ryan [“Applications of topological tensor products to infinite dimensional holomorphy”, Thesis, Trinity Coll., Dublin (1980; per bibl.)] had a strong influence. A detailed study of the relation between the norm of a polynomial and the norm of its associated symmetric multilinear form is presented in Chapter 1. Isometric and isomorphic properties of Banach spaces of polynomials are analyzed. Recent advances in Grothendieck’s problem of topologies on the bounded subsets of the projective tensor product of two Fréchet spaces, due to Taskinen and others, are relevant in Section 1.2. The theme of Chapter 2 is the duality and preduality of spaces of polynomials. Different types of polynomials are introduced.
Chapter 3 examines the basic properties of holomorphic functions defined on a locally convex space. Monomial and Taylor expansions are discussed. Both play an important role in the next chapters. The convergence of these expansions is investigated. Different types of decompositions of a space are introduced. The application of decompositions to obtain expansions of holomorphic mappings constitutes the main tool in Chapter 4. While Chapter 3 concentrates on properties common to Banach spaces and nuclear spaces, Chapter 4 tries to integrate the theories of holomorphic functions on Banach spaces and on nuclear spaces in a unified theory of holomorphic functions on Fréchet spaces. The coincidence of two topologies on spaces of holomorphic functions on different types of spaces is thoroughly studied. The necessity of several hypotheses which appear in the positive results in Sections 4.2 and 4.3 is examined; very interesting examples and counterexamples are presented in Section 4.4.
The comparison of locally convex properties of spaces of holomorphic functions in Chapters 3 and 4 relies on expansions which converge everywhere, at least pointwise, to the functions under consideration. This confines the investigation to balanced domains. To treat arbitrary open sets, ideas and methods of a nonlinear kind which originated in the theory of several complex variables must be introduced. The space of germs of holomorphic functions on an arbitrary compact subset is defined in Chapter 5, Section 5.1. Riemann domains over locally convex spaces are discussed in the next section. The main results presented in Section 5.2 are the relationship between a Riemann domain and its envelope of holomorphy, the Oka-Weil approximation theorem, and the solution to the Levi problem. Some of the main results are due to Schottenloher and Mujica.
Chapter 6 deals with the extension of holomorphic mappings. In Section 6.1 the author studies the extension of holomorphic functions defined on a dense subspace of a locally convex space; this section includes work by Meise, Vogt and the author. In Section 6.2 the extension of holomorphic functions defined on closed subspaces is studied. A nice account of the norm-preserving Aron-Berner extension of $$n$$-homogeneous polynomials to the bidual of a Banach space is given. Finally, Section 6.3 examines the spectrum of the Fréchet algebra $$H_b(E)$$ of entire functions of bounded type defined on a Banach space $$E$$.
Every chapter includes many exercises with different levels of difficulty. A separate appendix at the end of the book provides hints, comments, and cross-references about many of the exercises. Each chapter finishes with a long section with very interesting historical remarks and precise references. Nearly 900 items appear in the excellent list of references. Personal comments of the author about infinite holomorphy are also included in these notes. In fact this book was written by one of the main researchers in this area. He has made essential contributions to it. Moreover, he has a very attractive way of writing. There is no other book available which treats this topic so thoroughly, and with so much updated information.
The book reflects very well the interaction between (multi-)linear functional analysis and complex-analytic function theory. Despite the amount of information and the many references to recent work, the book is not only a reference tool. It is very readable and will be useful to readers who are interested not only in polynomials or infinite holomorphy, but also in Banach spaces, locally convex spaces, topological tensor products, complex function theory, several complex variables, topological algebras, etc. This important book culminates years of exhaustive research, and it will be indispensable for present and future researchers in this and related areas.

##### MSC:
 46G20 Infinite-dimensional holomorphy 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces