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Two refinements of the Bishop-Phelps-Bollob'as modulus | | Mario Chica
; Vladimir Kadets
; Miguel Martin
; Javier Meri
; Mariia Soloviova
; | | Date: |
21 Oct 2014 | | Abstract: | Extending the celebrated result by Bishop and Phelps that the set of norm
attaining functionals is always dense in the topological dual of a Banach
space, Bollob’as proved the nowadays known as the Bishop-Phelps-Bollob’as
theorem, which allows to approximate at the same time a functional and a vector
in which it almost attains the norm. Very recently, two
Bishop-Phelps-Bollob’as moduli of a Banach space have been introduced [J.
Math. Anal. Appl. 412 (2014), 697--719] to measure, for a given Banach space,
what is the best possible Bishop-Phelps-Bollob’as theorem in this space. In
this paper we present two refinements of the results of that paper. On the one
hand, we get a sharp general estimation of the Bishop-Phelps-Bollob’as modulus
as a function of the norms of the point and the functional, and we also
calculate it in some examples, including Hilbert spaces. On the other hand, we
relate the modulus of uniform non-squareness with the Bishop-Phelps-Bollob’as
modulus obtaining, in particular, a simpler and quantitative proof of the fact
that a uniformly non-square Banach space cannot have the maximum value of the
Bishop-Phelps-Bollob’as modulus. | | Source: | arXiv, 1410.5570 | | Services: | Forum | Review | PDF | Favorites | |
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