A modified Bishop-Phelps-Bollobas theorem and its sharpness

V. Kadets; M. Soloviova

doi:10.15330/ms.44.1.84-88

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 $\ell_\infty^{(2)}$.

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-Bollobґas moduli of a Banach space, J. Math. Anal. Appl., 412 (2014), №2, 697–719.

5. Chica M., Kadets V., Martin M., Meri J., Soloviova, M., Two refinements of the Bishop-Phelps-Bollobґas modulus, Banach J. Math. Anal., 9 (2015), №4, 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.

	A modified Bishop-Phelps-Bollobas theorem and its sharpness
Author	V. Kadets, M. Soloviova 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 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-Bollobґas moduli of a Banach space, J. Math. Anal. Appl., 412 (2014), №2, 697–719. 5. Chica M., Kadets V., Martin M., Meri J., Soloviova, M., Two refinements of the Bishop-Phelps-Bollobґas modulus, Banach J. Math. Anal., 9 (2015), №4, 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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