On norm attaining Bloch maps

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Given a complex Banach space X, let Bˆ(D,X) denote the space of all normalized Bloch maps from the open complex unit disc D into X. We prove that the set of all maps in Bˆ(D,X) which attain their Bloch norms is norm dense in Bˆ(D,X). Our approach is based on a previous study of the extremal structure of the unit closed ball of G(D) (the Bloch-free Banach space over D). We prove that normalized Bloch atoms of D are precisely the only extreme points of that ball and, in fact, they are strongly exposed points. Moreover, we characterize the surjective linear isometries of G(D) involved the Möbius transformations of D.