Series expansions in Fréchet spaces and their duals; construction of Fréchet frames

Abstract

Frames for Fréchet spaces $X_F$ with respect to Fréchet sequence spaces $Θ_F$ are studied and conditions, implying series expansions in $X_F$ and $X_F^*$, are determined. If $\{g_i\}$ is a $Θ_0$-frame for $X_0$, we construct a sequence $\{X_s\}_{s\in {\mathbb N}_0}$, $X_s\subset X_{s-1}$, $s\in {\mathbb N}$, for given $Θ_F$, respectively a sequence $\{Θ_s\}_{s\in {\mathbb N}_0}$, $Θ_s\subset Θ_{s-1}$, $s\in {\mathbb N}$, for given $X_F$, so that $\{g_i\}$ is a pre-$F$-frame (or $F$-frame) for $X_F$ with respect to $Θ_F$ under different assumptions given on $X_0$, $Θ_0$, $Θ_F$ or $X_0$, $Θ_0$, $X_F$.