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Intersection properties of ball sequences and uniqueness of Hahn–Banach extensions

Published online by Cambridge University Press:  14 November 2011

E. Oja  and
M. Põldvere
E. Oja
Affiliation:
Faculty of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia ([email protected], [email protected])
M. Põldvere
Affiliation:
Faculty of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia ([email protected])
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Extract

Let X be a Banach space and Y a closed subspace. We introduce an intrinsic geometric property of Y—the k-ball sequence property—which is a weakening of the famous k-ball property due to Alfsen & Effros. We prove that Y satisfies the 2-ball sequence property if and only if Y has the Phelps uniqueness property U (i.e. every continuous linear functional g ∈Y* has a unique norm-preserving extension f ∈X*). We prove that Y is an ideal having property U if and only if Y satisfies the 3-ball sequence property, and in this case, Y satisfies the k-ball sequence property for all k.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1251 - 1262
Copyright
Copyright © Royal Society of Edinburgh 1999

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