关于Banach空间的光滑性(G-可微)与强光滑性(F-可微)

SMOOTHNESS AND STRONG SMOOTHNESS OF BANACH SPACES

设X为Banach空间。文中证明了X的范数在x_0∈S(X)={x∈X,‖x‖=1}Gateax可微(相应地,Fréchet可微)的充分必要条件为:对任何y∈X■(相应地,■)其中,X_Ⅱ=span{x_0,y}。

In this paper, it is demonstrated that for a Banach space X, the norm of X is Gateuxdifferentiable (respectively, Fréchet differentiable) at x_0∈S(X){x∈X, ‖x‖=1} if andonly if the following condition holds■