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THE BISHOP-PHELPS-BOLLOBÁS VERSION OF LINDENSTRAUSS PROPERTIES A AND B.
- Source :
- Transactions of the American Mathematical Society; Sep2015, Vol. 367 Issue 9, p6085-6101, 17p
- Publication Year :
- 2015
-
Abstract
- We study a Bishop-Phelps-Bollob´as version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y) has the Bishop-Phelps-Bollob´as property (BPBp) for every Banach space Y. We show that in this case, there exists a universal function ηX(ε) such that for every Y, the pair (X, Y) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y) has the Bishop-Phelps-Bollob´as property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollob´as property for c<subscript>0</subscript>-, ℓ<subscript>1</subscript>- and ℓ∞-sums of Banach spaces. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 367
- Issue :
- 9
- Database :
- Complementary Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 103289317
- Full Text :
- https://doi.org/10.1090/S0002-9947-2015-06551-9