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The conjugate of a smooth Banach space

Published online by Cambridge University Press:  17 April 2009

D. G. Tacon
D. G. Tacon
Affiliation:
Australian National University, Canberra, ACT.
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Abstract

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A Banach space X is smooth if at every point of the unit sphere there is only one supporting hyperplane of the unit ball; and strictly convex, or rotund, if the unit sphere contains no line segment.

Although there is a strong duality between these notions, Klee has produced a smooth space whose conjugate is not rotund. However there is no known example of a smooth space with conjugate not isomorphic to a rotund space.

The main purpose of this note is to show that if X is a smooth space with a certain property, X* is isomorphic to a rotund space. This will follow from a mapping theorem which implies the existence of a set Γ and a continuous one-to-one linear map T of X* into co(Γ).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 3 , June 1970 , pp. 415 - 425
Copyright
Copyright © Australian Mathematical Society 1970

References

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