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Dentability indices and locally uniformly convex renormings. (English) Zbl 0801.46010
Summary: We prove that if the dentability index $$\delta (X)$$ of a Banach space $$X$$ is less than $$\omega_ 1$$ (first uncountable ordinal), then $$X$$ admits an equivalent locally uniformly convex norm. We prove also that if its weak$$^*$$ dentability index $$\delta^* (X)$$ is less than $$\omega_ 1$$, then $$X$$ admits an equivalent norm whose dual norm is locally uniformly convex.

##### MSC:
 46B22 Radon-Nikodým, Kreĭn-Milman and related properties 46B03 Isomorphic theory (including renorming) of Banach spaces
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##### References:
 [1] R.D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodym property , Lecture Notes in Math. 993 (1983). · Zbl 0512.46017 [2] R. Deville, Problèmes de renormages , J. Funct. Anal. 68 (1986), 117-129. · Zbl 0607.46014 [3] C.A. Edgar and R.F. Wheeler, Topological properties of Banach spaces , Pacific J. Math. 115 (1984). · Zbl 0506.46007 [4] R. Haydon, A counterexample to several questions about scattered compact spaces , Bull. London Math. Soc. 22 (1990), 261-268. · Zbl 0725.46007 [5] ——–, Trees in renorming theory , · Zbl 1036.46003 [6] R. Haydon and C.A. Rogers, A locally uniformly convex renorming for certain $$C(K)$$ , Mathematika 37 (1990), 1-8. · Zbl 0725.46008 [7] R.C. James, Some self dual properties of normed linear spaces , Symposium on Infinite Dimensional Topology, Ann. Math. Stud. 69 (1972), 159-175. · Zbl 0233.46025 [8] ——–, A separable somewhat reflexive Banach space with non separable dual , Bull. Amer. Math. Soc. 80 (1974), 738-743. · Zbl 0286.46018 [9] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $$l_1$$ and whose duals are non separable , Studia Math. 54 (1975), 81-105. · Zbl 0324.46017 [10] E. Odell and H.P. Rosenthal, A double dual characterization of separable Banach spaces containing $$l_1$$ , Israel J. Math. 20 (1975). · Zbl 0312.46031 [11] W. Szlenk, The nonexistence of a separable reflexive Banach space universal for all separable reflexive Banach spaces , Studia Math. 30 (1968), 53-61. · Zbl 0169.15303 [12] M. Talagrand, Renormages de quelques $$C(K)$$ , Israel J. Math. 54 (1986), 327-334. · Zbl 0611.46023
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