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# Uniform Convexity and Bishop-Phelps-Bollobás Property

Published:2013-04-02
Printed: Apr 2014
• Sun Kwang Kim,
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
• Han Ju Lee,
Department of Mathematics Education, Dongguk University - Seoul, 100-715 Seoul, Republic of Korea
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## Abstract

A new characterization of the uniform convexity of Banach space is obtained in the sense of Bishop-Phelps-Bollobás theorem. It is also proved that the couple of Banach spaces $(X,Y)$ has the bishop-phelps-bollobás property for every banach space $y$ when $X$ is uniformly convex. As a corollary, we show that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on $\ell_p\times \ell_q$ ($1\lt p, q\lt \infty$).
 Keywords: Bishop-Phelps-Bollobás property, Bishop-Phelps-Bollobás theorem, norm attaining, uniformly convex
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 46B22 - Radon-Nikod{y}m, Kreiin-Milman and related properties [See also 46G10]