Approximations of holomorphic functions in certain Banach spaces

Let E be a Banach space and let B(R) subset of E denote the open ball with centre at 0 and radius R. The following problem is studied: given 0 < r < R, epsilon > 0 and a function f holomorphic on B(R), does there always exist an entire function g on E such that \f - g\ < epsilon on B(r)? L. Lempert has proved that the answer is positive for Banach spaces having an unconditional basis with unconditional constant 1. In this paper a somewhat shorter proof of Lemperts result is given. In general it is not possible to approximate f by polynomials since f does not need to be bounded on B(r).