Series expansions in Fréchet spaces and their duals, construction of Fréchet frames

Frames for Fréchet spaces XFXF with respect to Fréchet sequence spaces ΘFΘF are studied, and conditions implying series expansions in XFXF and XF∗ are determined. If {gi}i=1∞ is a Θ0Θ0-frame for X0X0 and ΘFΘF (resp. XFXF) is given, we construct a sequence {Xs}s∈N0{Xs}s∈N0, Xs⊂Xs−1Xs⊂Xs−1, s∈Ns∈N, (resp. {Θs}s∈N0{Θs}s∈N0, Θs⊂Θs−1Θs⊂Θs−1, s∈Ns∈N), so that {gi}i=1∞ is a pre-FF-frame or FF-frame for XFXF with respect to ΘFΘF under different assumptions given on X0X0, Θ0Θ0 and ΘFΘF (resp. XFXF).