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- W1520008814 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." @default.
- W1520008814 created "2016-06-24" @default.
- W1520008814 creator A5005236475 @default.
- W1520008814 date "2012-03-22" @default.
- W1520008814 modified "2023-09-23" @default.
- W1520008814 title "Topological properties of two-modular convergence" @default.
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