The absolutely representing families in certain classes of locally convex spaces

A collection $X_\Lambda=\{x_\alpha\colon\alpha\in\Lambda\}$ of nonzero elements of a complete separable locally convex space $H$ over a field of scalars $\Psi$ ($\Psi=\mathbb R$ or $\mathbb C$), where $\Lambda$ is a certain set of indices, is said to be an absolutely representing family (ARF) in $H$ if $\forall x\in H$ one can find a family in the form $\{c_\alpha x_\alpha\colon c_\alpha\in\Psi$, $\alpha\in\Lambda\}$, that is absolutely summable to $x$ in $H$. In this paper we study certain properties of ARFs in the Fréchet spaces and strong adjoints to reflexive Fréchet spaces. We pay the most attention to obtaining the criteria that allow one to conclude that a given collection $X_\Lambda$ is an ARF in $H$.