Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Differentiation of real-valued functions and continuity of metric projections

Author: Simon Fitzpatrick
Journal: Proc. Amer. Math. Soc. 91 (1984), 544-548
MSC: Primary 46G05; Secondary 49A52, 58C20, 58C25
MathSciNet review: 746087
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We characterize the Fréchet differentiability of real-valued functions on certain real Banach spaces in terms of a directional derivative being equal to a modified version of the local Lipschitz constant. This yields the continuity of metric projections onto closed sets whose distance functions have directional derivatives equal to 1, provided the Banach space and its dual have Fréchet differentiable norms.

References [Enhancements On Off] (What's this?)

  • [1] Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
  • [2] Simon Fitzpatrick, Metric projections and the differentiability of distance functions, Bull. Austral. Math. Soc. 22 (1980), no. 2, 291–312. MR 598702, 10.1017/S0004972700006596
  • [3] John R. Giles, Convex analysis with application in the differentiation of convex functions, Research Notes in Mathematics, vol. 58, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 650456
  • [4] R. R. Phelps, Differentiability of convex functions on Banach spaces, lecture notes from a postgraduate course, University College, London, 1978.
  • [5] V. L. Šmulian, Sur la structure de la sphère unitaire dans l’espace de Banach, Rec. Math. [Mat. Sbornik] N.S. 9 (51) (1941), 545–561 (French). MR 0005775
  • [6] L. P. Vlasov, Almost convex and Čebyšev sets, Mat. Zametki 8 (1970), 545–550 (Russian). MR 0276736

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46G05, 49A52, 58C20, 58C25

Retrieve articles in all journals with MSC: 46G05, 49A52, 58C20, 58C25

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society