A geometry characteristic of Banach spaces with c~1-norm

摘要

Let E be a Banach space with the c~1-norm ‖•‖ in E{0}, and let S(E) = {e ∈ 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{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‖•‖in E{0} and differential structure of S(E).