Complex Analysis on Infinite Dimensional Spaces

Infinite dimensional holomorphy is the study of holomorphic or analytic func tions over complex topological vector spaces. The terms in this description are easily stated and explained and allow the subject to project itself ini tially, and innocently, as a compact theory with well defined boundaries. However, a comprehensive study would include delving into, and interacting with, not only the obvious topics of topology, several complex variables theory and functional analysis but also, differential geometry, Jordan algebras, Lie groups, operator theory, logic, differential equations and fixed point theory. This diversity leads to a dynamic synthesis of ideas and to an appreciation of a remarkable feature of mathematics - its unity. Unity requires synthesis while synthesis leads to unity. It is necessary to stand back every so often, to take an overall look at one's subject and ask "How has it developed over the last ten, twenty, fifty years? Where is it going? What am I doing?" I was asking these questions during the spring of 1993 as I prepared a short course to be given at Universidade Federal do Rio de Janeiro during the following July. The abundance of suit able material made the selection of topics difficult. For some time I hesitated between two very different aspects of infinite dimensional holomorphy, the geometric-algebraic theory associated with bounded symmetric domains and Jordan triple systems and the topological theory which forms the subject of the present book.

This text is unique - there are no other texts available on this subject It is written by an expert in the field and author of "The Schwarz Lemma" (now out of print)

Klappentext

This book considers basic questions connected with, and arising from, the locally convex space structures that may be placed on the space of holomorphic functions over a locally convex space. The first three chapters introduce the basic properties of polynomials and holomorphic functions over locally convex spaces. These are followed by two chapters concentrating on relationships between the compact open topology, the ported or Nachbin topology and the topology generated by the countable open covers. The concluding chapter examines the interplay between the various concepts introduced earlier as being intrinsic to infinite dimensional holomorphy. The comprehensive notes, historical background, exercises, appendix and bibliography make this book an invaluable reference whilst the presentation and synthesis of ideas from different areas will appeal to mathematicians from many different backgrounds.

Inhalt

1. Polynomial.- 1.1 Continuous Polynomials.- 1.2 Topologies on Spaces of Polynomials.- 1.3 Geometry of Spaces of Polynomials.- 1.4 Exercises.- 1.5 Notes.- 2. Duality Theory for Polynomial.- 2.1 Special Spaces of Polynomials and the Approximation Property.- 2.2 Nuclear Spaces.- 2.3 Integral Polynomials and the Radon-Nikodým Property.- 2.4 Reflexivity and Related Concepts.- 2.5 Exercises.- 2.6 Notes.- 3. Holomorphic Mappings between Locally Convex Space.- 3.1 Holomorphic Functions.- 3.2 Topologies on Spaces of Holomorphic Mappings.- 3.3 The Quasi-Local Theory of Holomorphic Functions.- 3.4 Polynomials in the Quasi-Local Theory.- 3.5 Exercises.- 3.6 Notes.- 4. Decompositions of Holomorphic Function.- 4.1 Decompositions of Spaces of Holomorphic Functions.- 4.2 ?? ?? for Fréchet Spaces.- 4.3 ?b ?? for Fréchet Spaces.- 4.4 Examples and Counterexamples.- 4.5 Exercises.- 4.6 Notes.- 5. Riemann Domain.- 5.1 Holomorphic Germs on a Fréchet Space.- 5.2 Riemann Domains over Locally Convex Spaces.- 5.3 Exercises.- 5.4 Notes.- 6. Holomorphic Extension.- 6.1 Extensions from Dense Subspaces.- 6.2 Extensions from Closed Subspaces.- 6.3 Holomorphic Functions of Bounded Type.- 6.4 Exercises.- 6.5 Notes.- Appendix. Remarks on Selected Exercises.- References.