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On projectional resolution of identity on the duals of certain Banach spaces

Published online by Cambridge University Press:  17 April 2009

M. Fabian
M. Fabian
Affiliation:
sibeliova 49, Prague 6, Czechoslovakia
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Abstract

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A consequence of the main proposition includes results of Tacon, and John and Zizler and says: If a Banach space X possesses a continuous Gâteaux differentiable function with bounded nonempty support and with norm-weak continuous derivative, then its dual X* admits a projectional resolution of the identity and a continuous linear one-to-one mapping into c0 (Γ). The proof is easy and selfcontained and does not use any complicated geometrical lemma. If the space X is in addition weakly countably determined, then X* has an equivalent dual locally uniformly rotund norm. It is also shown that l∞ admits no continuous Gâteaux differentiable function with bounded nonempty support.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 35 , Issue 3 , June 1987 , pp. 363 - 371
Copyright
Copyright © Australian Mathematical Society 1987

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