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A Non-Reflexive Banach Space has Non-Smooth Third Conjugate Space

Published online by Cambridge University Press:  20 November 2018

J. R. Giles
J. R. Giles*
Affiliation:
The University Of Newcastle, N.S.W. 2308, Australia
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J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 117 - 119
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Bishop, Errett and Phelps, R. R., A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc, 67 (1961), 97-98.Google Scholar
2. Bollobás, Béla, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc, 2 (1970), 181-182.Google Scholar
3. Dixmier, J., Sur un théorème de Banach, Duke Math. J., 15 (1948), 1057–1071.Google Scholar
4. Giles, J. R., On a characterisation of differentiability of the norm of a normed linear space, J. Austral. Math. Soc, 12 (1971), 106-114.Google Scholar
5. Giles, J. R., On a differentiability condition for reflexivity of a Banach space, J. Austral. Math. Soc, 12 (1971), 393-396.Google Scholar
6. James, R. C., A characterisation of reflexivity, Studia Math., 23 (1964), 205-216.Google Scholar
7. Wilansky, A., Functional Analysis, Blaisdell 1964.Google Scholar
8. Day, M. M., Normed Linear Spaces, 3rd ed. Springer 1973.Google Scholar
9. Phelps, R. R., A representation theorem for bounded convex sets, Proc Amer. Math. Soc 11 (1960), 976-983.Google Scholar