VECTOR SPACES OF ENTIRE FUNCTIONS OF UNBOUNDED TYPE

Let E be an infinite dimensional complex Banach space. We prove the existence of an infinitely generated algebra, an infinite dimensional closed subspace and a dense subspace of entire functions on E whose non-zero elements are functions of unbounded type. We also show that the tau delta topology on the space of all holomorphic functions cannot be obtained as a countable inductive limit of Frechet spaces.