Recent results on analytic mappings and plurisubharmonic functions in topological linear spaces

Abstract

In t roduc t ion In the past four years, results were obtained to extend complex analysis to topological vector spaces, which are not Banach spaces. A good motivation is to solve problems in functional spaces, mostly Fr~chet spaces. We give here (see sections 4, 5 of this lecture) examples, concerning the space A(~) of the analytic functions in a domain ~C ~n , endowed with the topology of compact convergence, and spaces of entire functions. Analytic mappings, plurisubharmonic functions and the notions of polar and negligible sets are introduced through classical problems in functional spaces. We give also a generalization of the classical Banach-Steinhaus theorem for complex vector spaces and polynomial mappings, which leads us to introduce the notion of C-barrelled space. For the proofs of the basic results concerning plurisubharmonic functions, the reader is referred to the book [12] and to my seminar lectures [14] and [15], to the Note [13] and to the theses of Ph. Noverraz [21] and G. Coeur~ [4].