Baire trees, bad norms and the Namioka property

Abstract

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.