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Generic Fréchet differentiability of convex operators


Author: Nikolai K. Kirov
Journal: Proc. Amer. Math. Soc. 94 (1985), 97-102
MSC: Primary 46G05; Secondary 47H99, 90C25
DOI: https://doi.org/10.1090/S0002-9939-1985-0781064-4
MathSciNet review: 781064
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Abstract: We consider order-bounded convex operators $ F:E \to X$ from a reflexive Banach space $ E$ into a Banach lattice $ X$. In both cases (i) $ X$ and $ {X^*}$ have weak compact intervals, and (ii) $ X$ has norm compact intervals, we obtain that $ F$ is Fréchet differentiable at the points of some dense $ {G_\delta }$ subset of $ E$.


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  • [1] K. T. Andrews, The Radon-Nikodym property for spaces of operators, J. London Math. Soc. 28 (1983), 113-122. MR 703470 (85d:47047)
  • [2] J. M. Borwein, Continuity and differentiability properties of convex operators, Proc. London Math. Soc. 44 (1982), 420-444. MR 656244 (83h:58011)
  • [3] -, Subgraadients of convex operators (preprint).
  • [4] A. V. Buchvalov, A. I. Veksler and G. Ya. Losanowski, Banach lattices--some Banach aspects of the theory, Uspehi Mat. Nauk 34 (1979), 137-183. MR 535711 (80f:46019)
  • [5] J. P. R. Christensen, Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings, Proc. Amer. Math. Soc. 86 (1982), 649-655. MR 674099 (83k:54014)
  • [6] J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, vol. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
  • [7] S. Heinrich, N. J. Nielsen and G. H. Olsen, Order bounded operators and tensor products of Banach lattices, Math. Scand. 49 (1981), 99-127. MR 645092 (84a:46154)
  • [8] N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267-278. MR 0341154 (49:5904)
  • [9] P. S. Kenderov, Dense strong continuity of pointwise continuous mappings, Pacific J. Math. 89 (1980), 111-130. MR 596921 (82d:46034)
  • [10] -, Monotone operators in Asplund spaces, C. R. Acad. Bulgare Sci. 30 (1977), 963-965. MR 0463981 (57:3919)
  • [11] -, Semicontinuity of set-valued monotone mappings, Fund. Math. 88 (1975), 61-69. MR 0380723 (52:1620)
  • [12] N. K. Kirov, Differentiability of convex mappings and generalized monotone mappings, C. R. Acad. Bulgare Sci. 34 (1981), 1473-1475. MR 654429 (83i:46047)
  • [13] -, Generalized monotone mappings and differentiability of vector-valued convex mappings, Serdica 9 (1983), 263-274. MR 744156 (85h:46069)
  • [14] I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), 735-750. MR 0390721 (52:11544)
  • [15] R. R. Phelps, Differentiabilty of convex functions on Banach spaces, Lecture Notes, Univ. College, London, 1977.
  • [16] C. Stegall, The duality between Asplund spaces and spaces with Radon-Nikodym property, Israel J. Math. 29 (1978), 408-412. MR 0493268 (58:12297)
  • [17] M. Valadier, Sous-differentiabilité des fonctions convexes a valeurs dans un espace vectoriel ordonné, Math. Scand. 30 (1972), 65-74. MR 0346525 (49:11250)

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DOI: https://doi.org/10.1090/S0002-9939-1985-0781064-4
Article copyright: © Copyright 1985 American Mathematical Society

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