Approximation of holomorphic functions in Banach spaces admitting a Schauder decomposition

Let X be a complex Banach space. Recall that X admits a finite-dimensional Schauder decomposition if there exists a sequence {X-n}(n=1) (infinity) of finite-dimensional subspaces of X, such that every X E X has a unique representation of the form x = Sigma(infinity)(n=1) x(n), with X-n is an element of X-n for every n. The finite-dimensional Schauder decomposition is said to be unconditional if, for every X E X, the series x = Sigma(infinity)(n=1) n(x), which represents x, converges unconditionally, that is, Sigma(infinity)(n=1) x(pi) (n) converges for every permutation pi of the integers. For short, we say that X admits an unconditional F.D.D. We show that if X admits an unconditional F.D.D. then the following Runge approximation property holds: (R.A.P.) There is r is an element of (0, 1) such that, for any epsilon > 0 and any holomorphic function f on the open unit ball of X, there exists a holomorphic function h on X satisfying \f - h\ < epsilon on the open ball of radius r centered at the origin.