Extension and Lifting Theorems for Analytic Mappings

Abstract

Publisher Summary This chapter discusses the extension and lifting theorems for analytic mappings. Some results concerning lifting and extension theorems for analytic mappings between complex Banach spaces are presented. The necessary and sufficient condition on complex Banach spaces is found so that for each function, a suitably rich class of holomorphic mapping, there is an analytic extension. The theorem suffers from the defect that it only treats very special pairs for which analytic extensions are not only possible but are also linear and continuous. There is very rarely a linear continuous extension mapping. These special situations occur when some kind of nuclearity condition is present. A Banach space has the bounded approximation property if for some constant can be approximated uniformly on compact, convex, balanced subsets by elements of norm at most. It is found that for any complex Banach space, locally convex spaces are topologically isomorphic to the Frechet space of all the functions.