关于Banach空间中连续凸函数的Gateaux可微性的一个注记

通过引进了范—ω＊一致连续的定义，指出了若Ｂａｎａｃｈ空间Ｘ中每个开凸子集Ｄ上定义的连续凸函数ｆ（ｘ）的次微分在Ｄ的一个稠集上范—ω＊一致连续，则ｆ（ｘ）一定在Ｄ的一个稠Ｇδ集上Ｇａｔｅａｕｘ可微。

On the Extension of Isometries between Unit Spheres of E and C（Ω）

In this paper, we study the extension of isometries between the unit spheresof some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S_1(E) of all smoothpoints of the unit sphere S_1(E) is dense in S_1(E), then under some condition, every surjectiveisometry V_0 from S_1(E) onto S_1(C(Ω)) can be extended to be a real linearly isometric map V of Eonto C(Ω). From this result we also obtain some corollaries. This is the first time we study thisproblem on different typical spaces, and the method of proof is also very different too.