A geometry characteristic of Banach spaces with c (1)-norm

作者：Ma Jipu^*

来源：Frontiers of Mathematics in China, 2014, 9(5): 1089-1103.

DOI：10.1007/s11464-014-0385-3

• Banach space
• geometry
• non-linear analysis
• global analysis

Let E be a Banach space with the c (1)-norm aEuro-center dot aEuro- in E\{0}, and let S(E) = {e a E: aEuro-eaEuro- = 1}. In this paper, a geometry characteristic for E is presented by using a geometrical construct of S(E). That is, the following theorem holds: the norm of E is of c (1) in E\{0} if and only if S(E) is a c (1) submanifold of E, with codimS(E) = 1. The theorem is very clear, however, its proof is non-trivial, which shows an intrinsic connection between the continuous differentiability of the norm aEuro-center dot aEuro- in E\{0} and differential structure of S(E).

• 出版日期2014-10
• 单位南京大学