关于粗范数的注记

A NOTE ON ROUGH NORMS

设(X,q)为实Banach空间,S(X)为(X,q)中的单位球面,对于x∈S(X),记集合{f∈S(X):f(x)=q(x)}为A(x),我们获得了下述结果:若存在正数ε使对于x∈S(X)及δ>0.y∈S(X)满足q(y-x)< δ且dian[A(x)∪A(y)≥ε;则对于x∈X,sup z∈S(X)[q(x+z)+q(x-z)]≥2q(x)+ε,这里正数ε恰如前述,作为推论,我们证明了:范数为粗当且仅当它为一致不Frechet可微,从而回答了[1]中所提出的问题。

A Let(X.q)he a Banach space and S(X)the unit sphere in (X.q). For any x∈S(X),the set {f∈S(X*):f(x)=q(x)} is denoted byA(x) .We prove the following result; If there exists ε>0 such that for any x6S(X) and any δ>0,there is y∈S(X) satisfying q(y-x)<s and diam[A(x≥∪ A(y)]≥ε :then for any x∈X, sup [q(x + z) +q(x-z)]≥2q(x)+ε,where e is just as indicated above. From this.it follows that a norm is rough if and only if it is uniformly non-Frcchet differentiable. Thus a question raised in [1] is answered in the affirmative.