Certain properties of Generalization of $L^p-$Spaces for $0<p<1$

Abstract

This paper introduces the notion of $N^{\ast }-$function and gives a generalization of $L^p,$ for $0<p<1$ denoted by $L_{\Phi }$ where $\Phi$ is an $N^{\ast }-$function. As well as, this paper examines some properties regarding to this generalized spaces and its linear forms, including some analogies and common features to some other well known spaces. As well as, we prove this space is a quasi-normed space but it is not normed space.