An analogue of Liapounoff's convexity theorem for Birnbaum-Orlicz spaces and the extreme points of their unit balls (Q1068322)

For the Birnbaum-Orlicz-spaces \(L_{\Phi}(\mu)\) over a non-atomic probability measure \(\mu\) the Young-functions involved satisfying the \(\Delta\) \({}_ 2\)-condition, the author studies the extreme points of the unit ball B of \(L_{\Phi}(\mu)\) with respect to the Luxemburg- and the Orlicz- (or functional-) norm. In particular he obtains that T(ext B)\(=T(B)\) for every continuous linear \(T: L_{\Phi}(\mu)\to {\mathbb{R}}^ n\) which is an analogue of Liapounoff's convexity theorem considered as a result on \(\sigma (L^{\infty},L^ 1)\)-continuous \(T: L_{\infty}(\mu)\to {\mathbb{R}}^ n\).