Dual renormings of Banach spaces

We prove that a Banach space admitting an equivalent WUR norm is an Asplund space. Some related dual renormings are also presented. It is a well-known result that a Banach space whose dual norm is Fréchet differentiable is reflexive. Also if the the third dual norm is Gâteaux differentiable the space is reflexive. For these results see e.g. [2], p.33. Similarly, by the result of [9], if the second dual norm is Gâteaux differentiable the space is an Asplund space. The main result of this note answers a question posed by Troyanski. It claims that a space that admits an equivalent weakly uniformly rotund (WUR) norm is an Asplund space. By the well known dual characterization of WUR norms (see [1]) it is equivalent to the dual norm being uniformly Gâteaux differentiable (UG). In fact, the existence of an equivalent (not necessarily dual) UG norm on the dual space is sufficient. This follows from the dual characterization of UG norms as norms the dual of which is weak-star uniformly rotund (WUR). The restriction of a WUR norm of the second dual space to the original space is easily shown to be WUR. Let us point out that by our Theorem 4 merely dual Gâteaux norm does not imply, in general, that the space is Asplund. In the remaining part of the paper we strive to improve our knowledge of the higher dual norms of separable spaces. We show that the space J of James admits a dual norm that is WUR. In particular, spaces whose second dual norm is UG do not necessarily have to be reflexive. Let us recall that by a classical result [8] duals of separable spaces containing an isomorphic copy of l1 contain an isomorphic copy of l1(c). As a consequence, these duals do not have an equivalent Gâteaux smooth renorming. Therefore the second dual norm of these spaces cannot be rotund. In contrast, spaces with separable dual admit a WUR norm whose second dual is WUR. We investigate the existence of the second dual rotund norm on the classical James tree space (JT) and Hagler space (JH). These spaces do not contain an isomorphic copy of l1, but their dual is nonseparable. In Theorem 4 we construct an equivalent norm on the James tree space whose second dual is uniformly rotund in every direction ( URED). On the other hand we prove that the space JH of Hagler has no equivalent norm whose second dual is rotund. Altogether, the class of separable Banach spaces that admit a norm whose second dual is rotund lies strictly between spaces with separable dual and spaces not containing an isomorphic copy of l1. In line of these results it seems natural to ask whether duals of separable Banach In part supported by NSERC (Canada). AMS Classification: 46B03,46B20.