Best Simultaneous Approximation in L^p(I,E)

Let G be a reflexive subspace of the Banach space E, and L^p(I,E)(1≤p<∞) denote the space of all p-Bochner integrable functions on I=0,1 with values in E. Norm N(·,…,·) is developed on R ^m(m≥2), where N is nondecreasing in each coordinateon of the set R ₊^m. It is shown that L^p(I,G) is an N-simultaneously approximation in L^p(I,E).