Abstract of a paper by Jipu Ma
This is an announcement for the paper "A geometry characteristic for Banach space with $c^1$-norm" by Jipu Ma. Abstract: Let $E$ be a Banach space with the $c^1$-norm $\|\cdot\|$ in $ E \backslash \{0\}$ and $S(E)=\{e\in E: \|e\|=1\}.$ 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 $c^1$ in $ E \backslash \{0\}$ if and only if $S(E)$ is a $c^1$-submanifold of $E,$ with ${\rm codim}S(E)=1.$ The theorem is very clear, however, its proof is non-trivial, which shows an intrinsic connection between the continuous differentiability of the norm $\|\cdot\|$ in $ E \backslash \{0\}$ and differential structure of $S(E).$ Archive classification: math.FA Mathematics Subject Classification: 54Exx, 46Txx, 58B20 Submitted from: huangql@yzu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.6823 or http://arXiv.org/abs/1109.6823
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