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**Complex analysis in Banach spaces. Holomorphic functions and domains of holomorphy in finite and infinite dimensions.**
*(English)*
Zbl 0586.46040

North-Holland Mathematics Studies, 120. Notas de MatemĂˇtica, 107. Amsterdam/New York/Oxford: North-Holland. XI, 434 p. $ 55.25; Dfl. 160.00 (1986).

It is well known that problems arising from the study of holomorphic continuation and approximation have been central in the development of complex analysis in finitely many variables. In this book the author presents a unified view of these topics both in finite and in infinite dimension. Based on a course given in Campinas this book gives a complete, clear and self-contained exposition of the main results obtained in Banach spaces as well as in \({\mathbb{C}}^ n.\)

The first part is devoted to the basic definitions and properties of holomorphic mappings and domains of holomorphy. The second part deals with differential forms and the \({\bar \partial}\)-operator. Polynomially convex domains and commutative Banach algebras are investigated.

The third part introduces plurisubharmonic functions and pseudo-convex domains. A chapter is devoted to the Levi problem and to the solution of the \({\bar \partial}\)-equation in pseudoconvex domains.

In the last part the preceding results are extended in the natural setting of Riemann domains.

The first part is devoted to the basic definitions and properties of holomorphic mappings and domains of holomorphy. The second part deals with differential forms and the \({\bar \partial}\)-operator. Polynomially convex domains and commutative Banach algebras are investigated.

The third part introduces plurisubharmonic functions and pseudo-convex domains. A chapter is devoted to the Levi problem and to the solution of the \({\bar \partial}\)-equation in pseudoconvex domains.

In the last part the preceding results are extended in the natural setting of Riemann domains.

Reviewer: Ph.Noverraz

### MSC:

46G20 | Infinite-dimensional holomorphy |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

32D05 | Domains of holomorphy |

32U05 | Plurisubharmonic functions and generalizations |