On sequence spaces for Fr\'echet frames

We analyze the construction of a sequence space $\tilde{\mathrm{\Theta }}$, resp. a sequence of sequence spaces, in order to have $\{g_i\}$ as a $\tilde{\mathrm{\Theta }}$-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.