Bounded Holomorphic Functions Attaining their Norms in the Bidual

  • Daniel Carando

    Universidad de Buenos Aires, Argentina
  • Martin Mazzitelli

    Universidad de Buenos Aires, Argentina

Abstract

Under certain hypotheses on the Banach space XX, we prove that the set of analytic functions in Au(X)\mathcal A_u (X) (the algebra of all holomorphic and uniformly continuous functions in the ball of XX) whose Aron–Berner extensions attain their norms is dense in Au(X)\mathcal A_u (X). This Lindenstrauss type result also holds for functions with values in a dual space or in a Banach space with the so-called property ( β\beta). We show that the Bishop–Phelps theorem does not hold for Au(c0,Z)\mathcal A_u (c_0, Z'') for a certain Banach space ZZ, while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.

Cite this article

Daniel Carando, Martin Mazzitelli, Bounded Holomorphic Functions Attaining their Norms in the Bidual. Publ. Res. Inst. Math. Sci. 51 (2015), no. 3, pp. 489–512

DOI 10.4171/PRIMS/162