On Fréchet spaces with certain classes of proximal subspaces

A new metric with absolutely convex balls is introduced on a metrizable locally convex space. Necessary and sufficient conditions are given for all closed hypersubspaces and all nonnormable closed subspaces of a Fréchet space to be proximal, i.e., to have the property that there exist elements of best approximation with respect to this metric. In particular, these conditions are expressed in terms of the topologies of the original space and the strong dual space. It is proved that the Fréchet spaces $B\times\omega$ have the proximality property, where $B$ is a reflexive Banach space and $\omega=R^N$ is the nuclear Fréchet space of all numerical sequences. Questions of Albinus and Wriedt are answered.
Bibliography: 23 titles.