Holomorphic maps of uniform type

Abstract

(E,X)to hold. This problem for vector-valued holomorphic maps, i.e. for the casewhere X is a locally convex space, was investigated by some authors. Thefirst result on this problem belongs to Colombeau and Mujica. In [2] theyhave shown that the equality (UN) holds when E is a dual Fr´echet–Montelspace and X a Fr´echet space. Next, a necessary and sufficient conditionfor (UN) to hold in the class of scalar holomorphic functions on a nuclearFr´echet space was established by Meise and Vogt [7]. An important sufficientcondition for (UN) for scalar holomorphic functions on such a space was alsofound recently by those two authors [8]. However, until now, when X doesnot have a linear structure, the problem has not been investigated.Here we consider this problem for holomorphic maps with values in acomplex manifold of infinite dimension, in particular, in the projective spaceassociated with a Fr´echet space (see the definition in §2). In the first section,by the method of [4], we give a characterization of the uniformity of holo-morphic maps with values in complex Banach manifolds. The scalar casehas been proved by Meise and Vogt [7] by a different method. Section 2is devoted to proving the main result (Theorem 2.1) of this note: everyholomorphic map from a dual space of a nuclear Fr´echet space (i.e., from