A geometry characteristic of Banach spaces with c¹-norm

Abstract

Let E be a Banach space with the c¹-norm || ∙ || in E\{0}, and let S(E) = {e\inE: ||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).

Keywords

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

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg