  This paper introduces the notion of $N^*-$function and gives a generalization
of $L^p,$ for $0<p<1$ denoted by $L_\Phi$ where $\Phi$ is an $N^*-$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.
