A generalization of the Koshi-Shimogaki quasi-norm and a topology in the modular spaces in the sense of Nakano (Q1820357)

The note contains very general formula for F-quasi-norms in the modular space generated by a (u,v)-convex modular \(\rho\) (or in particular (s,t)- convex modular) on a vector space X with a non-trivial valuation. The author presented properties of the norms \(\| \cdot \|^ A_{\rho}\), \(\| \cdot \|_{\rho}\), \(\| \cdot \|^ A_{s,t}\) and their connections with Amemiya's norm and Koshi-Shimogaki quasi-norm. The last part of the note contains a very interesting remark that the linear topology on X generated by an F-quasi norm has a neighborhood base of zero (consisting of sets which are defined in the note).