ℂ-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation. (6th July 2006) - Filipsson Filipsson, Lars Lars
Abstract : We investigate the concepts of linear convexity and ℂ -convexity in complex Banach spaces. The main result is that any ℂ -convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a ℂ -convex domain Ω in the Banach space X and a point p ∉ Ω, there is a complex hyperplane through p that does not intersect Ω . We also prove that linearly convex domains are holomorphically convex, and that Kergin interpolation can be performed on holomorphic mappings defined in ℂ -convex domains.