Sharp corner points and isometric extension problem in Banach spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:405

Article

Ding, Guang-Gui^1,2  Li, Jian-Ze^2,3,4 

[1] Nankai Univ, Sch Math Sci, Tianjin 300071, Peoples R China

[2] Nankai Univ, LPMC, Tianjin 300071, Peoples R China

[3] Nankai Univ, Chern Inst Math, Tianjin 300071, Peoples R China

[4] Tianjin Univ, Sch Sci, Tianjin 300072, Peoples R China

关键词: Isometric extension;    Sharp corner point;    Weak*-exposed point;    Weak-Asplund space;    Asplund generated space;    Weakly compactly generated space;    Gateaux differentiable space;   

DOI  :  10.1016/j.jmaa.2013.04.002

来源: Elsevier

【 摘 要 】

In this article, we begin using some geometric methods to study the isometric extension problem in general real Banach spaces. For any Banach space Y. we define a collection of sharp corner points of the unit ball B-1(Y*), which is empty if Y is strictly convex and dim Y >= 2. Then we prove that any surjective isometry between two unit spheres of Banach spaces X and Y has a linear isometric extension on the whole space if Y is a gateaux differentiability space (in particular, separable spaces or reflexive spaces) and the intersection of sharp corner points and weak*-exposed points of B(Y*) is weak*-dense in the latter. Moreover, we study the sharp corner points in many classical Banach spaces and solve isometric extension problem affirmatively in the case that Y is (l(1)), c(0)(Gamma), c(Gamma), l(infinity) (Gamma) or some C(Omega). (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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