Topological properties of two-modular convergence

Ewa Kasior

Abstract

Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure space and assume that E is an ideal of $L_0$. Let $\rho$ and $\rho^*$ be two modulars defined on $E, E_\rho$ and $E_{\rho^*}$ the modular spaces for $\rho$ and $\rho^*$ respectively. In $E_\rho \cap E_{\rho^*}$ a two-modular convergence can be defined as follows: a sequence $(f_n)$ in $E_\rho \cap E_{\rho^*}$ is said to be two-modular convergent to $f \in E_\rho \cap E_{\rho^*}$ whenever $f_n \rightarrow f$ with respect to modular $\rho^*$ and $(f_n)$ is $\rho$-bounded. We introduce a two-modular topology $\gamma w (\mathcal{T}_\rho,\mathcal{T}_{\rho^*})$ in $E_\rho \cap E_{\rho^*}$ and show that the convergence in this topology is equivalent to the two-modular convergence. We prove also that the two-modular convergence is equivalent to some modular convergence. The most important fact on this paper is a characterization of linear functionals on the space $L^{\varphi 1}\cap L^{\varphi 2}$, continuous with respect to the two-modular topology $\gamma w (\mathcal{T}_{m_\rho 1},\mathcal{T}__{m_\rho 2})$. The functions $\varphi_1$ and $\varphi_2$ are not assumed to be convex.