Quasinormed spaces generated by a quasimodular

In this paper, we introduce the notion of a quasimodular and we prove that the respective Minkowski functional of the unit quasimodular ball becomes a quasinorm. 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 the basic properties of quasinormed Calderón–Lozanovskiĭ spaces $E_{\phi }$ $E_{\varphi}$ , where the lower Matuszewska–Orlicz index ${\alpha }_{\phi }$ $\alpha _{\varphi}$ plays the key role. Our studies are conducted in a full possible generality.