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 ${\mathrm{\Theta }}_F$ are studied and conditions, implying series expansions in $X_F$ and $X_F^{\ast }$, are determined. If $\{g_i\}$ is a ${\mathrm{\Theta }}_0$-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 ${\mathrm{\Theta }}_F$, respectively a sequence $\{{\mathrm{\Theta }}_s{\}}_{s\in {\mathbb{N}}_0}$, ${\mathrm{\Theta }}_s\subset {\mathrm{\Theta }}_{s-1}$, $s\in {\mathbb N}$,forgiven$X_F$,sothat${g_i}$isapre-$F$-frame(or$F$-frame)for$X_F$withrespectto$\Theta_F$ under different assumptions given on $X_0$, ${\mathrm{\Theta }}_0$, ${\mathrm{\Theta }}_F$ or $X_0$, ${\mathrm{\Theta }}_0$, $X_F$.