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Keywords:
strong subdifferentiability of norms; Asplund spaces; renormings; weak compact generating
strong subdifferentiability of norms; Asplund spaces; renormings; weak compact generating
Summary:
The strong subdifferentiability of norms (i.e\. one-sided differentiability uniform in directions) is studied in connection with some structural properties of Banach spaces. It is shown that every separable Banach space with nonseparable dual admits a norm that is nowhere strongly subdifferentiable except at the origin. On the other hand, every Banach space with a strongly subdifferentiable norm is Asplund.
References:
[1] Contreras M.D., Payá R.: On upper semicontinuity of duality mapping. Proc. Amer. Math. Soc. 121 (1994), 451-459. MR 1215199
[2] Deville R., Godefroy G., Zizler V.: Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics 64 (1993). MR 1211634 | Zbl 0782.46019
[3] Deville R., Godefroy G., Hare D., Zizler V.: Differentiability of convex functions and the convex point of continuity property in Banach spaces. Israel J. Math. 59 (1987), 245-255. MR 0920087 | Zbl 0654.46021
[4] Franchetti C.: Lipschitz maps and the geometry of the unit ball in normed spaces. Archiv. Math. 46 (1986), 76-84. MR 0829819 | Zbl 0564.46014
[5] Franchetti C., Payá R.: Banach spaces with strongly subdifferentiable norm. Boll. Uni. Mat. Ital. VII-B (1993), 45-70. MR 1216708
[6] Godefroy G.: Some applications of Simons' inequality. Seminar of Functional Analysis II, Univ. of Murcia, to appear. MR 1767034
[7] Godefroy G., Kalton N.J.: The ball topology and its applications. Contemporary Math. 85 (1989), 195-238. MR 0983386 | Zbl 0676.46003
[8] Gregory D.A.: Upper semicontinuity of subdifferential mappings. Canad. Math. Bull. 23 (1980), 11-19. MR 0573553
[9] Haydon R.: Trees in renorming theory. to appear. MR 1674838 | Zbl 1036.46003
[10] James R.C.: Weakly compact sets. Trans. Amer. Math. Soc. 113 (1964), 129-140. MR 0165344 | Zbl 0129.07901
[11] Jayne J.E., Rogers C.A.: Borel selectors for upper semicontinuous set valued mappings. Acta Math. 155 (1985), 41-79. MR 0793237
[12] John K., Zizler V.: Smoothness and its equivalents in weakly compactly generated Banach spaces. J. Funct. Anal. 15 (1974), 161-166. MR 0417759 | Zbl 0272.46012
[13] John K., Zizler V.: Projections in dual weakly compactly generated Banach spaces. Studia Math. 49 (1973), 41-50. MR 0336295 | Zbl 0247.46029
[14] Lindenstrauss J., Tzafriri L.: Classical Banach spaces I. Sequence Spaces Springer-Verlag (1977). MR 0500056 | Zbl 0362.46013
[15] Odell E., Rosenthal H.P.: A double dual characterization of separable Banach spaces containing $\ell_1$. Israel J. Math. 20 (1975), 375-384. MR 0377482 | Zbl 0312.46031
[16] Simons S.: A convergence theorem with boundary. Pacific J. Math. 40 (1972), 703-708. MR 0312193 | Zbl 0237.46012
[17] Singer I.: Bases in Banach Spaces II. Springer-Verlag (1981). MR 0610799 | Zbl 0467.46020
[18] Stegall C.: The Radon-Nikodym property in conjugate Banach spaces. Trans. Amer. Math. Soc. 206 (1975), 213-223. MR 0374381 | Zbl 0318.46056
[19] Troyanski S.L.: On a property of the norm which is close to local uniform convexity. Math. Ann. 27 (1985), 305-313. MR 0783556
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