Quasi-modular spaces with applications to quasi-normed Calder\'on-Lozanovski\u{\i} spaces

TL;DR

This paper introduces quasi-modulars and demonstrates their connection to quasi-norms, applying these concepts to analyze properties of quasi-normed Calderón-Lozanovski spaces and their structural embeddings.

Contribution

It develops the theory of quasi-modulars, extending classical modular theory to quasi-normed spaces, and applies this to Calderón-Lozanovski spaces with new results on their structure.

Findings

Quasi-modulars induce quasi-norms via Minkowski functional.

Characterization of $l^{ abla}$ copies in $E_{\varphi}$ spaces.

Results hold in full generality for broad classes of spaces.

Abstract

In this paper we introduce the notion of a quasi-modular and we prove that the respective Minkowski functional of the unit quasi-modular ball becomes a quasi-norm. In this way, we refer to and complete the well-known theory related to the notions of a modular and a convex modular that lead to the F-norm and to the norm, respectively. We use the obtained results to consider basic properties of quasi-normed Calder\'on-Lozanovski\u{\i} spaces $E_{\varphi}$, where the lower Matuszewska-Orlicz index $\alpha_{\varphi}$ plays the key role. We also give a number of theorems concerning different copies of $l^{\infty}$ in the spaces $E_{\varphi}$ in the natural language of suitable properties of the space $E$ and the function $\varphi$. Our studies are conducted in a full possible generality.