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Smoothability, Strong Smoothability and Dentability in Banach Spaces1

Published online by Cambridge University Press:  20 November 2018

R. Anantharaman ,
T. Lewis  and
J. H. M. Whitfield
R. Anantharaman
Affiliation:
Suny College at old Westbury, Old WestburyLong Island, New York
T. Lewis
Affiliation:
University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
J. H. M. Whitfield
Affiliation:
Lakehead University, Thunder Bay, Ontario, Canada., P7B 5E1
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Abstract

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It is shown that dentability of the unit ball of a conjugate Banach space X* does not imply smoothability of the unit ball of X, answering a question raised by Kemp. A property called strong smoothability is introduced and is shown to be dual to dentability. The results are used to provide new proofs of the facts that X is an Asplund space whenever it has an equivalent Fréchet differentiable norm, or whenever X* has the Radon-Nikodym Property.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 24 , Issue 1 , 01 March 1981 , pp. 59 - 68
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Asplund, E., Fréchet differentiability of convex functions, Acta Math. 121 (1962), 31-47.Google Scholar
2. Asplund, E. and Rockafellar, R. T., Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1962), 443-467.Google Scholar
3. Cox, S. H. Jr and Nadler, S. B. Jr, Supremum norm differentiability, Ann. Soc. Math. Polonae. 15 (1962), 127-131.Google Scholar
4. Diestel, J. and Faires, B., On vector measures, Trans. Amer. Math. Soc. 198 (1962), 253-271.Google Scholar
5. Diestel, J. and Uhl, J. J. Jr, The Radon-Nikodym Theorem for Banach space valued measures, Rocky Mountain J. Math. 6 (1962), 1-46.Google Scholar
6. Dunford, N. and Schwartz, J. T., Linear Operators I, Interscience (New York), 1958.Google Scholar
7. Edelstein, M., Smoothability versus dentability, Comment. Math. Univ. Carolinae. 14 (1962), 127-133.Google Scholar
8. Ekeland, I. and Lebourg, G., Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1962), 193-216.Google Scholar
9. Hocking, J. G. and Young, G. S., Topology, Addison-Wesley (Reading, Mass.), 1961.Google Scholar
10. Kemp, D. C., A note on smoothability in Banach spaces, Math. Ann. 218 (1962), 211-217.Google Scholar
11. Leonard, I. E. and Sundaresan, K., Smoothness and Duality in Lp (E, jut), J. Math. Anal. App. 46 (1962), 513-522.Google Scholar
12. Lewis, T., On the duality between smoothability and dentability, Proc. Amer. Math. Soc. 63 (1962), 239-244.Google Scholar
13. Namioka, I. and Phelps, R. R., Banach spaces which are Asplund spaces, Duke Math. J. 42 (1962), 735-750.Google Scholar
14. Phelps, R. R., Support cones in Banach spaces and their applications, Adv. in Math. 13 (1962), 1-19.Google Scholar
15. Phelps, R. R., Dentability and extreme points in Banach spaces, J. Func. Analysis. 16 (1962), 78-90.Google Scholar
16. Phelps, R. R., The duality between Asplund spaces and Radon-Nikodym spaces, Rainwater Seminar Notes, 1977.Google Scholar
17. Stegall, C., The Radon-Nikodým property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1962), 213-223.Google Scholar
18. Stegall, C., The duality between Asplund spaces and spaces with the Radon-Nikodym Property. Israel J. Math. 20 (1962), 408-412.Google Scholar
19. Sullivan, F., Dentability, smoothability, and stronger properties in Banach spaces, Indiana Univ. Math. J. 26 (1962), 545-553.10.1512/iumj.1977.26.26042Google Scholar
20. Sullivan, F., On the duality between Asplund spaces and spaces with the Radon-Nikodym Property. Proc. Amer. Math. Soc. 71 (1962), 155-156.Google Scholar