  We analyze the construction of a sequence space $\widetilde{\Theta}$, resp. a
sequence of sequence spaces, in order to have $\{g_i\}$ as a
$\widetilde{\Theta}$-frame or Banach frame for a Banach space $X$, resp.
pre-$F$-frame or $F$-frame for a Fr\'echet space $X_F=\cap_{s\in {\mathbb N}_0}
X_s$, where $\{X_s\}_{s\in {\mathbb N}_0}$ is a sequence of Banach spaces.
