On holomorphic functions attaining their norms

We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon-Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k, it cannot be approximated by norm attaining polynomials with degree less than k. For X = d(*) (omega, 1), a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm. (C) 2004 Elsevier Inc. All rights reserved.