对偶空间上凸函数的逼近

Approximation of Convex Functions on the Dual Spaces

设 Banach空间 E具有等价二次严格凸范数 ,f为其对偶空间 E*上的 w*-下半连续 Lips-chitz凸函数 ,该文证明了 E*上存在 w* -下半连续且很光滑点集稠密 (从而在稠子集上 Gateaux可微 )的 Lipschitz凸函数的单调序列 {fn}在有界集上一致逼近 f.

In this paper, the authors prove that for every \$w\+*\$ lower semicontinuous Lipschitzian convex function on the dual of a bistrictly convexifiable Banach space can be uniformly approximated by a sequence of \$w\+*\$ lower semicontinuous monotone nondecreasing Lipschitzian convex function with the dense very smooth point set.