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On the canonical projection of the third dual of a Banach space onto the first dual

Published online by Cambridge University Press:  17 April 2009

A.L. Brown
A.L. Brown
Affiliation:
School of Mathematics, University of Newcastle-upon-Tyne, Newcastle, England; Department of Mathematics, University of Newcastle, Newcastle, New South Wales.
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Abstract

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For a Banach space B let P denote the canonical projection of the third dual space of B onto the embedding of the first dual into the third. It is shown that if B = l1 then ‖I-P‖ = 2.

This fact shows to be mistaken a current belief in a statement which is equivalent to the statement that for all Banach spaces B the operator I – P is of norm one.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 3 , December 1976 , pp. 351 - 354
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Day, Mahlon M., Normed linear spaces, 3rd ed. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 21. Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[2]Dixmier, J., “Sur un théorème de Banach”, Duke Math. J. 15 (1948), 1057–1071.CrossRefGoogle Scholar