A modified BishopPhelpsBollobas theorem and its sharpness 

Author 
v.kateds@karazin.ua; 5_11_16@mail.ru
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 BishopPhelpsBollob\'{a}s theorem, there exists $(y,y^*)\in S_X \times X^*$ such that $\{y}^*\={y}^*({y})$, and
$ \max\{\xy\, \x^*y^*\\} \leq \sqrt{\varepsilon}$.
We show that this theorem is sharp in a number of twodimensional spaces, which makes a big difference with the original BishopPhelpsBollob\'{a}s theorem, where the only (up to isometry) twodimensional space, in which the theorem is sharp, is $\ell_\infty^{(2)}$.

Keywords 
BishopPhelpsBollobas theorem; norm attaining functional

DOI 
doi:10.15330/ms.44.1.8488

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 BishopPhelpsBollobas type theorem for uniform algebras, Advances in Mathematics. 240 (2013), 370–382. 4. Chica M., Kadets V., Martin M., MorenoPulido S., RamblaBarreno F., BishopPhelpsBollob´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 BishopPhelpsBollob´as modulus, Banach J. Math. Anal., 9 (2015), ¹4, 296–315. 6. Diestel J., Geometry of Banach spaces. – Lecture notes in Math., V.485, SpringerVerlag, 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, SpringerVerlag, Berlin, 1993. 
Pages 
8488

Volume 
44

Issue 
1

Year 
2015

Journal 
Matematychni Studii

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