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Separable determination of Fréchet differentiability of convex functions. (English) Zbl 0839.46036
It is shown that a continuous convex function defined on an open convex subset of a Banach space \(E\) and for which the subdifferential mapping has separable image on every closed separable subspace of \(E\), is Fréchet differentiable on a dense \(G_\delta\) subset of its domain. This sheds some light on Asplund spaces, which are defined as those Banach spaces, where every continuous convex function defined on an open subset is Fréchet differentiable on a dense \(G_\delta\)-subset, and which can be equally characterized by the property that every separable subspace has a separable dual [see R. R. Phelps, ‘Convex functions, monotone operators and differentiability’, Lect. Notes Math. 1364 (1989; Zbl 0658.46035)].
Reviewer: A.Kriegl (Wien)

46G05 Derivatives of functions in infinite-dimensional spaces
46B20 Geometry and structure of normed linear spaces
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
Full Text: DOI
[1] DOI: 10.1006/jmaa.1994.1466 · Zbl 0897.46025
[2] Giles, Bull. Austral. Math. Soc. 41 pp 371– (1990)
[3] Giles, J. Austral. Math. Soc. 32 pp 134– (1982)
[4] DOI: 10.1007/BF02787266 · Zbl 0654.46021
[5] Holmes, Geometrical functional analysis and its applications 24 (1975)
[6] Phelps, Convex functions, monotone operators and differentiability (1993) · Zbl 0921.46039
[7] Moors, Set Valued Analysis
[8] Kenderov, C.R. Acad. Bulgare Sci. 30 pp 963– (1977)
[9] DOI: 10.1090/S0002-9904-1974-13580-9 · Zbl 0286.46018
[10] Zajíček, Czechoslovak Math. J. 41 pp 471– (1991)
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