Holomorphic maps of discs into F-spaces

Abstract

At the Conference on In f in i te Dimensional Ho lomorphy held at the Univers i ty o f Ken tucky in 1973, D. J. Patil asked whether there exist ho lomorph ic maps f o f the open unit disc U into the open uni t ball B o f any separable complex 'Banach space, such that the convex hull o f f (U) is dense in B. The fo l lowing theorem shows that one can actual ly do much better (or worse, depending on one's a t t i tude toward pathology) ; in part icular, convex i ty plays no role.