Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-09T15:29:54.405Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Characterisation of Differentiability of the Norm of a Normed Linear Space

Published online by Cambridge University Press:  09 April 2009

J. R. Giles
J. R. Giles
Affiliation:
The University of Newcastle, N.S.W.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1971 , pp. 106 - 114
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Bishop, E. and Phelps, R. R., ‘A proof that every Banach space is subreflexive’, Bull. Amer. Math. Soc. 67 (1961), 97–98.Google Scholar
[2]Bonsall, F. F., Cain, B. E. and Schneider, H., ‘The numerical range of a continuous mapping of a normed space’, Aeq. Math. 2 (1968), 86–93.Google Scholar
[3]Cudia, D. F., ‘The geometry of Banach spaces. Smoothness’, Trans. Amer. Math. Soc. 110 (1964), 284–314.Google Scholar
[4]Nakano, H., ‘Topology and Linear Topological Spaces’ (Maruzen, Tokyo, 1951).Google Scholar
[5]Phelps, R. R., ‘A representation theorem for bounded convex sets’, Proc. Amer. Math. Soc. 11 (1960), 976–983.Google Scholar
[6]Šmulian, V. L., ‘On some geometrical properties of the unit sphere in the space of type (B)’, Mat. Sb. N.S. 48 (1938), 90–94.Google Scholar
[7]Šmulian, V. L., ‘Sur la dérivabilité de la norme dans l'espace de Banach’, Dokl. Akad. Nauk. SSSR 27 (1940), 643–648.Google Scholar
[8]Šmulian, V. L., ‘Sur la structure de la sphère unitaire dans l'espace de Banach’, Mat. Sb. N.S. 51 (1941), 545–561.Google Scholar
[9]Wilansky, A., ‘Functional Analysis’ (Blaisdell, New York, 1964).Google Scholar