On upper semicontinuity of duality mappings

Contreras, Manuel D.; Payá, Rafael

doi:10.1090/S0002-9939-1994-1215199-4

On upper semicontinuity of duality mappings
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by Manuel D. Contreras and Rafael Payá PDF

Proc. Amer. Math. Soc. 121 (1994), 451-459 Request permission

Abstract:

We give new sufficient conditions for a Banach space to be an Asplund (or reflexive) space in terms of certain upper semicontinuity of the duality mapping.

References

C. Aparicio, F. Ocaña, R. Payá, and A. Rodríguez, A nonsmooth extension of Fréchet differentiability of the norm with applications to numerical ranges, Glasgow Math. J. 28 (1986), no. 2, 121–137. MR 848419, DOI 10.1017/S0017089500006443
Edgar Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31–47. MR 231199, DOI 10.1007/BF02391908
Dennis F. Cudia, The geometry of Banach spaces. Smoothness, Trans. Amer. Math. Soc. 110 (1964), 284–314. MR 163143, DOI 10.1090/S0002-9947-1964-0163143-9
Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
J. Diestel and B. Faires, On vector measures, Trans. Amer. Math. Soc. 198 (1974), 253–271. MR 350420, DOI 10.1090/S0002-9947-1974-0350420-8
Ivar Ekeland and Gérard Lebourg, Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), no. 2, 193–216 (1977). MR 431253, DOI 10.1090/S0002-9947-1976-0431253-2
Ky Fan and Irving Glicksberg, Some geometric properties of the spheres in a normed linear space, Duke Math. J. 25 (1958), 553–568. MR 98976

Duality mapping and homeomorphisms in Banach spaces

Carlo Franchetti and Rafael Payá, Banach spaces with strongly subdifferentiable norm, Boll. Un. Mat. Ital. B (7) 7 (1993), no. 1, 45–70 (English, with Italian summary). MR 1216708
J. R. Giles, D. A. Gregory, and Brailey Sims, Geometrical implications of upper semi-continuity of the duality mapping on a Banach space, Pacific J. Math. 79 (1978), no. 1, 99–109. MR 526669
Gilles Godefroy, Application à la dualité d’une propriété d’intersection de boules, Math. Z. 182 (1983), no. 2, 233–236 (French). MR 689300, DOI 10.1007/BF01175625

Some applications of Simons’ inequality

G. Godefroy and N. J. Kalton, The ball topology and its applications, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 195–237. MR 983386, DOI 10.1090/conm/085/983386
Gilles Godefroy and Václav Zizler, Roughness properties of norms on non-Asplund spaces, Michigan Math. J. 38 (1991), no. 3, 461–466. MR 1116501, DOI 10.1307/mmj/1029004394
David A. Gregory, Upper semicontinuity of subdifferential mappings, Canad. Math. Bull. 23 (1980), no. 1, 11–19. MR 573553, DOI 10.4153/CMB-1980-002-9
Richard Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (1990), no. 3, 261–268. MR 1041141, DOI 10.1112/blms/22.3.261
Zhibao Hu and Bor-Luh Lin, Smoothness and the asymptotic-norming properties of Banach spaces, Bull. Austral. Math. Soc. 45 (1992), no. 2, 285–296. MR 1155487, DOI 10.1017/S000497270003015X
Robert R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR 984602, DOI 10.1007/BFb0089089
S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703–708. MR 312193
Charles Stegall, The duality between Asplund spaces and spaces with the Radon-Nikodým property, Israel J. Math. 29 (1978), no. 4, 408–412. MR 493268, DOI 10.1007/BF02761178
Wen Yao Zhang, On weakly very smooth Banach spaces, Northeast. Math. J. 3 (1987), no. 2, 140–142 (Chinese, with English summary). MR 952679