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A Non-Reflexive Smooth Space with a Smooth Dual

Published online by Cambridge University Press:  20 November 2018

J. H. M. Whitfield
J. H. M. Whitfield*
Affiliation:
Lakehead University
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Let (E, ρ) and (E*ρ*) be a real Banach space and its dual. Restrepo has shown in [4] that, if p and ρ* are both Fréchet differentiable, E is reflexive. The purpose of this note is to show that Fréchet differentiability cannot be replaced by Gateaux differentiability. This answers negatively a question raised by Wulbert [5]. In particular, we will renorm a certain nonreflexive space with a smooth norm whose dual is also smooth.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 4 , 01 December 1974 , pp. 579 - 580
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Asplund, E., Averaged norms, Israel J. Math. 5 (1967), 227-233.Google Scholar
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3. Phelps, R.R., A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976-983.Google Scholar
4. Restrepo, G., Differentiate norms, Soc. Mat. Mexicana Bol. 10 (1965), 47-55.Google Scholar
5. Wulbert, D., Approximation by Ck-functions, Proc. Sympos. on Approx. Theory Austin, 1973, Academic Press (to appear).Google Scholar