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Uniform Convexity and the Bishop–Phelps–Bollobás Property

Published online by Cambridge University Press:  20 November 2018

Sun Kwang Kim  and
Han Ju Lee
Sun Kwang Kim
Affiliation:
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun–gu, Seoul 130- 722, Republic of Korea. e-mail: lineksk@gmail.com
Han Ju Lee
Affiliation:
Department of Mathematics Education, Dongguk University–Seoul, 100-715 Seoul, Republic of Korea. e-mail: hanjulee@dongguk.edu
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Abstract

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A new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop–Phelps–Bollobás theorem. It is also proved that the couple of Banach spaces $\left( X,Y \right)$ 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}}$$\left( 1\,<p,q\,<\,\infty \right)$.

Keywords

46B2046B22Bishop–Phelps–Bollobás propertyBishop–Phelps–Bollobás theoremnorm attaininguniformly convex
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 2 , 01 April 2014 , pp. 373 - 386
Copyright
Copyright © Canadian Mathematical Society 2014

References

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