On modular spaces over a field with valuation (Q1093865)

In this note X denotes a vector space over a field K with nontrivial valuation. The author considered k-quasi-convex modulars on the space X. The main result of the note says that if the modular \(\rho\) satisfies the following two conditions: (1) \(\rho (x_ n)\to 0\) implies \(\rho (\alpha x_ n)\to 0\) for every \(\alpha\in K,\) (2) for every \(x\in X\) there exists \(y\in X\) such that \(2\rho\) (y)\(\leq \rho (x)\) and \(2\rho\) (x-y)\(\leq \rho (x),\) and if the valuation is nonarchimedean, then there exists a non nontrivial continuous linear functional on the modular space \(X_{\rho}\).