Frontiers of Mathematics in China >
A geometry characteristic of Banach spaces with c1-norm
Received date: 04 Feb 2010
Accepted date: 26 Feb 2014
Published date: 26 Aug 2014
Copyright
Let E be a Banach space with the c1-norm || ∙ || in E\{0}, and let S(E) = {eE: ||e|| = 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 c1 in E\{0} if and only if S(E) is a c1 submanifold of E, with codim S(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 || ∙ || in E\{0} and differential structure of S(E).
Key words： Banach space; geometry; non-linear analysis; global analysis
Jipu MA . A geometry characteristic of Banach spaces with c1-norm[J]. Frontiers of Mathematics in China, 2014 , 9(5) : 1089 -1103 . DOI: 10.1007/s11464-014-0385-3
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