Basic sequences in the dual of a Frechet space

In this paper we study some properties of basic sequences in the dual of a Frechet space. As a consequence we obtain that if E is a Frechet space with the property that for each closed subspace F of E and each bounded subset B of E/F there is a bounded subset A of E with phi (A)=B, where phi denotes the canonical surjection of E onto E/F, then one of the following conditio ns is at least satisfied: 1. E is a Banach space, 2. E is a Schwartz space, 3. E is the product of a Banach space by omega. Finally, we also obtain so me results concerning totally reflexive spaces.