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

TL;DR

This paper generalizes the classical $L^p$ spaces for $0<p<1$ using $N^*-$functions, introduces $L_\Phi$ spaces, and explores their properties, including their quasi-normed structure and relation to other spaces.

Contribution

It introduces $L_\Phi$ spaces based on $N^*-$functions as a generalization of $L^p$ spaces for $0<p<1$, and analyzes their properties and linear forms.

Findings

$L_\Phi$ spaces are quasi-normed but not normed.

Established analogies between $L_\Phi$ and other classical spaces.

Proved properties of linear forms on $L_\Phi$ spaces.

Abstract

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.