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First Borel class sets in Banach spaces and the asymptotic-norming property. (English) Zbl 1065.46010
This article deals with Banach spaces which are Borel sets of the first class in their biduals equipped with the weak-star topology, and contains in particular extensions of results due to G. A. Edgar and R. F. Wheeler [Pac. J. Math. 115, 317–350 (1984; Zbl 0506.46007))] and N. A. Ghoussoub and B. Maurey [J. Funct. Anal. 61, No. 1, 72–97 (1985; Zbl 0565.46011)); Proc. Am. Math. Soc. 94, No. 4, 665–671 (1985; Zbl 0587.46015); Mem. Am. Math. Soc. 62, No. 349 (1986; Zbl 0606.46005)]. Edgar and Wheeler showed that when the complement of a Banach space $$X$$ in its bidual is a countable union of weak-star compact convex sets, then $$X$$ has the Radon–Nikodym property. The converse, however, fails. It is shown in the article under review that the equivalence holds true for separable spaces $$X$$ if one replaces “weak-star compact convex sets” by “differences of two weak-star compact convex sets”.
Moreover, this representation can be obtained with differences which are at positive distance (in norm) from $$X$$ if and only if there is an equivalent norm on $$X$$ with the asymptotically norming property, without assuming separability of $$X$$. Other interesting renorming results are proved: for instance, an arbitrary Banach space has an equivalent locally uniformly rotund norm if and only if every norm open subset is a countable union of differences of closed convex sets. Let us mention that it is apparently still unknown whether every Banach space with the Radon–Nikodym property has an equivalent locally uniformly rotund norm.

##### MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces 46B20 Geometry and structure of normed linear spaces 46B22 Radon-Nikodým, Kreĭn-Milman and related properties
##### Keywords:
 [1] A. Bellow,Lifting compact spaces, Lecture Notes in Mathematics794, Springer-Verlag, Berlin, 1980, pp. 233–253. [2] J. Bourgain and F. Delbaen, A special class ofL spaces, Acta Mathematica145 (1980), 155–176. · Zbl 0466.46024 [3] R. D. Bourgin,Geometric Aspects of Convex Sets with Radon-Nikodým Property, Lecture Notes in Mathematics993, Springer-Verlag, Berlin, 1980. [4] B. Cascales and G. Vera,Topologies weaker than the weak topology of a Banach space, Journal of Mathematical Analysis and Applications182 (1994), 41–68. · Zbl 0808.46021 [5] M. Cepedello-Boiso,Approximation of Lipschitz functions by {$$\Delta$$}-convex functions in Banach spaces, Israel Journal of Mathematics109 (1998), 269–284. · Zbl 0920.46010 [6] R. Deville, G. Godefroy and V. Zizler,Smoothness and Renorming in Banach Spaces, Pitman Monographs and Surveys 64, Longman Sci. Tech., Harlow, 1993. · Zbl 0782.46019 [7] G. A. Edgar,Measurability in a Banach space I, Indiana University Mathematical Journal26 (1977), 663–667. · Zbl 0361.46017 [8] G. A. Edgar and R. F. Wheeler,Topological properties of Banach spaces, Pacific Journal of Mathematics115 (1984), 317–350. · Zbl 0506.46007 [9] N. Ghoussoub and B. Maurey,H {$$\delta$$}-embeddings in Hilbert space, Journal of Functional Analysis61 (1984), 72–97. · Zbl 0565.46011 [10] N. Ghoussoub and B. Maurey,The asymptotic norming and the Radon-Nikodým properties are equivalent, Proceedings of the American Mathematical Society94 (1985), 665–671. · Zbl 0587.46015 [11] N. Ghoussoub and B. Maurey,H {$$\delta$$}-embeddings in Hilbert space and optimization onG {$$\delta$$}-sets, Memoirs of the American Mathematical Society349 (1986),. [12] Z. Hu and B.-L. Lin,On the asymptotic norming property of Banach spaces, Lecture Notes in Pure and Applied Mathematics136, Dekker, New York, 1992, pp. 195–210. · Zbl 0834.46007 [13] R. C. James and A. Ho,The asymptotic-norming and the Radon-Nikodým properties for Banach spaces, Arkiv der Matematik19 (1981), 53–70. · Zbl 0466.46025 [14] J. E. Jayne, I. Namioka and C. A. Rogers,Topological properties of Banach spaces, Proceedings of the London Mathematical Society (3)66 (1993), 651–672. · Zbl 0793.54026 [15] K. Kuratowski,Topology, Volume I, PWN Polish Scientific Publishers, Warsaw, 1966. [16] G. Lancien,Théorie de l’indice et problèmes de renormage en géométrie des espaces de Banach, Thèse, Paris, 1992. [17] G. Lancien,On the Szlenk index and the weak*-dentability index, The Quarterly Journal of Mathematics. Oxford Series (2)47 (1996), 59–71. · Zbl 0973.46014 [18] P. W. McCartney and R. C. O’Brien,A separable Banach space with the Radon-Nikodým property that is not isomorphic to a subspace of a separable dual, Proceedings of the American Mathematical Society78 (1980), 40–42. · Zbl 0393.46018 [19] A. Moltó, J. Orihuela and S. Troyanski,Locally uniform rotund renorming and fragmentability, Proceedings of the London Mathematical Society (3)75 (1997), 619–640. · Zbl 0909.46011 [20] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia,Kadec and Krein-Milman properties, Comptes Rendus de l’Académie des Sciences, Paris, Série I331 (2000), 459–464. · Zbl 0983.46012 [21] I. Namioka,Radon-Nikodým compact spaces and fragmentability, Mathematika34 (1989), 258–281. · Zbl 0654.46017 [22] M. Raja,Kadec norms and Borel sets in a Banach space, Studia Mathematica136 (1999), 1–16. · Zbl 0935.46021 [23] M. Raja,Locally uniformly rotund norms, Mathematika46 (1999), 343–358. · Zbl 1031.46022