On a dual problem. I. General results. Applications to Frèchet spaces

Let $H$ be a separated locally convex space; $x_k\in H$, $x_k\ne0$, $k=1,2,\dots$ . The author shows that if $H$ is a Frèchet space or an $LN^*$-space, then the system $\{x_k\}$ is a basis (topological or absolute) in the closure of its linear span if and only if the system of equations $\varphi(x_k)=d_k$, $k=1,2,\dots$, has a solution $\varphi$ in $H'$ for any sequence $\{d_k\}$ from a certain space $E_1$ (respectively, from $E_2$ for an absolute basis).
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