We use Simonenko quantitative indices of an -function to estimate two parameters and in Orlicz function spaces with Orlicz norm, and get the following inequality: , where and are Simonenko indices. A similar inequality is obtained in with Orlicz norm.
We establish some results that concern the Cauchy-Peano problem in Banach spaces. We first prove that a Banach space contains a nontrivial separable quotient iff its dual admits a weak*-transfinite Schauder frame. We then use this to recover some previous results on quotient spaces. In particular, by applying a recent result of Hájek-Johanis, we find a new perspective for proving the failure of the weak form of Peano's theorem in general Banach spaces. Next, we study a kind of algebraic genericity...
A slight modification of the definition of the Colombeau generalized functions allows to have a canonical embedding of the space of the distributions into the space of the generalized functions on a manifold. The previous attempt in [5] is corrected, several equivalent definitions are presented.