On sequence spaces for Fréchet frames

We analyze the construction of a sequence space $\widetildeΘ$, resp. a sequence of sequence spaces, in order to have $\{g_i\}$ as a $\widetildeΘ$-frame or Banach frame for a Banach space $X$, resp. pre-$F$-frame or $F$-frame for a Fréchet 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.

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