Gruenhage Compacta and Strictly Convex Dual Norms
نویسنده
چکیده
We prove that if K is a Gruenhage compact space then C (K) admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X∗ = span|||·|||(K), where K is a Gruenhage compact in the w∗-topology and ||| · ||| is equivalent to a coarser, w∗-lower semicontinuous norm on X∗, then X∗ admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if Υ is a tree, then C0(Υ) ∗ admits an equivalent, strictly convex dual norm if and only if Υ is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.
منابع مشابه
Trees, Linear Orders and Gâteaux Smooth Norms
We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a Gâteaux smooth norm on C0(Υ), where Υ is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fréchet smooth norms. In addition, we prove that if C0(Υ) admits a Gâteaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.
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