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Points of Weak-Norm Continuity in the Unit Ball of Banach Spaces

✍ Scribed by T.S.S.R.K. Rao

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
71 KB
Volume
265
Category
Article
ISSN
0022-247X

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✦ Synopsis

Using the M-structure theory, we show that several classical function spaces and spaces of operators on them fail to have points of weak-norm continuity for the identity map on the unit ball. This gives a unified approach to several results in the literature that establish the failure of strong geometric structure in the unit ball of classical function spaces. Spaces covered by our result include the Bloch spaces, dual of the Bergman space L 1 and spaces of operators on them, as well as the a Ε½ . space C T rA, where A is the disc algebra on the unit circle T. For any unit vector f in an infinite-dimensional function algebra A we explicitly construct a Γ„ 4 sequence f in the unit ball of A that converges weakly to f but not in the norm.

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