Series expansions in Fr\'echet spaces and their duals; construction of Fr\'echet frames

Frames for Fr\'echet spaces $X_F$ with respect to Fr\'echet sequence spaces $\Theta_F$ are studied and conditions, implying series expansions in $X_F$ and $X_F^*$, are determined. If $\{g_i\}$ is a $\Theta_0$-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 $\Theta_F$, respectively a sequence $\{\Theta_s\}_{s\in {\mathbb N}_0}$, $\Theta_s\subset \Theta_{s-1}$, $s\in {\mathbb N}$, for given $X_F$, so that $\{g_i\}$ is a pre-$F$-frame (or $F$-frame) for $X_F$ with respect to $\Theta_F$ under different assumptions given on $X_0$, $\Theta_0$, $\Theta_F$ or $X_0$, $\Theta_0$, $X_F$.