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Separable determination of Fréchet differentiability of convex functions

Published online by Cambridge University Press:  17 April 2009

J.R. Giles  and
Scott Sciffer
J.R. Giles
Affiliation:
Department of MathematicsThe University of NewcastleNewcastle NSW 2308
Scott Sciffer
Affiliation:
Department of MathematicsThe University of NewcastleNewcastle NSW 2308
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Abstract

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For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Fréchet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 1 , August 1995 , pp. 161 - 167
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Deville, R., Godefroy, G., Hare, D.E.G. and Zizler, V., ‘Differentiability of convex functions and the convex point of continuity property in Banach spaces’, Israel J. Math. 59 (1987), 245–255.Google Scholar
[2]Giles, J.R., ‘On the characterisation of Asplund spaces’, J. Austral. Math. Soc. Ser. A 32 (1982), 134–144.Google Scholar
[3]Giles, J.R. and Sciffer, Scott, ‘Continuity characterisations of differentiability of locally Lipschitz functions’, Bull. Austral. Math. Soc. 41 (1990), 371–380.Google Scholar
[4]Giles, J.R. and Sciffer, Scott, ‘Generalising generic differentiability properties from convex to locally Lipschitz functions’, J. Math. Anal. Appl. 188 (1994), 833–854.Google Scholar
[5]Holmes, R.B., Geometrical functional analysis and its applications, Graduate Texts in Mathematics 24 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[6]James, R.C., ‘A separable somewhat reflexive Banach space with nonseparable dual’, Bull. Amer. Math. Soc. 80 (1974), 738–743.Google Scholar
[7]Kenderov, P.S., ‘Monotone operators in Asplund spaces’, C.R. Acad. Bulgare Sci. 30 (1977), 963–964.Google Scholar
[8]Moors, W.B., ‘A characterisation of minimal subgradients of locally Lipschitz functions’, Set Valued Analysis (to appear).Google Scholar
[9]Phelps, R.R., Convex functions, monotone operators and differentiability, Lecture notes in mathematics 1364 2nd ed. (Springer-Verlag, Berlin, Heidelberg, New York, 1993).Google Scholar
[10]Veselý, L., ‘Norm to weak upper semi-continuous monotone operators are generically strongly continuous’, (unpublished manuscript).Google Scholar
[11]Zajíček, L., ‘Fréchet differentiability, strict differentiability and subdifferentiability’, Czechoslovak Math. J. 41 (1991), 471–489.Google Scholar