On the Range of Analytic Functions into a Banach Space

Abstract

Publisher Summary This chapter focuses on the range of analytic functions into a Banach space. It was found that the solution of Patil's problem in the finite dimensional case was an application of the generalization of the well-known Rudin-Carleson theorem. But in the infinite dimensional case Patil's problem was solved independently. As it is easier to construct the continuous functions having some prescribed range properties than the analytic ones, it is obvious that various generalizations of the Rudin-Carleson theorem are an efficient tool when proving the existence of an analytic function whose range has certain density properties. Any closed ball in a complex Banach space X has the analytic extension property; to prove the same for more general closed sets is more difficult.