A modified Bishop-Phelps-Bollobas theorem and its sharpness

Author
Department of Mathematics and Informatics, Kharkiv V.N. Karazin National University
Abstract
Let $X$ be a real Banach space, $\varepsilon \in (0, 1)$, and let $(x, x^*) \in S_X \times S_{X^*}$ with $x^*(x) \ge 1 - \varepsilon$. Then, according to the modified Bishop-Phelps-Bollob\'{a}s theorem, there exists $(y,y^*)\in S_X \times X^*$ such that $\|{y}^*\|={y}^*({y})$, and $\max\{\|x-y\|, \|x^*-y^*\|\} \leq \sqrt{\varepsilon}$. We show that this theorem is sharp in a number of two-dimensional spaces, which makes a big difference with the original Bishop-Phelps-Bollob\'{a}s theorem, where the only (up to isometry) two-dimensional space, in which the theorem is sharp, is $\ell_\infty^{(2)}$.
Keywords
Bishop-Phelps-Bollobas theorem; norm attaining functional
DOI
doi:10.15330/ms.44.1.84-88
Reference
1. Bishop E., Phelps R.R., A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67 (1961), 97–98.

2. Bollobas, B., An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181–182.

3. Cascales B., Kadets V., Guirao A.J., A Bishop-Phelps-Bollobas type theorem for uniform algebras, Advances in Mathematics. 240 (2013), 370–382.

4. Chica M., Kadets V., Martin M., Moreno-Pulido S., Rambla-Barreno F., Bishop-Phelps-Bollob´as moduli of a Banach space, J. Math. Anal. Appl., 412 (2014), ¹2, 697–719.

5. Chica M., Kadets V., Martin M., Meri J., Soloviova, M., Two refinements of the Bishop-Phelps-Bollob´as modulus, Banach J. Math. Anal., 9 (2015), ¹4, 296–315.

6. Diestel J., Geometry of Banach spaces. – Lecture notes in Math., V.485, Springer-Verlag, Berlin, 1975.

7. Phelps R.R., Support cones in Banach spaces and their applications, Adv. Math., 13 (1974), 1–19.

8. Phelps R.R., Convex functions, monotone operators and differentiability (second edition). – Lecture Notes in Math., V.1364, Springer-Verlag, Berlin, 1993.

Pages
84-88
Volume
44
Issue
1
Year
2015
Journal
Matematychni Studii
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