On the quasinormability of Hb(U).

Abstract

Let E be a complex locally convex space, U a balanced open subset of E. The author continues the study of the quasinormability of the space Hb(U) of all holomorphic functions on U which are bounded on the U-bounded subsets of U. His main goal is to prove the following theorem: "For all Fréchet spaces E with the (BB)∞-property, Hb(U) is quasinormable for every balanced open subset U of E''; this includes all the known results about the quasinormability of Hb(U) in the Fréchet space setting