Reflexivity and best approximations in Fréchet spaces

The paper gives a negative answer to the following question of M. Wriedt: Is it true that in every projective limit of reflexive Banach spaces there exists a normlike metric for which all closed hyperplanes are proximinal?
In particular, it is shown that if $E[\mathfrak T]$ is a nuclear Fréchet space nonisomorphic to the space of all sequences $\omega$, then for an arbitrary normlike metric $d$ on $E$ inducing the topology $\mathfrak T$, there exist nonproximinal closed hyperplanes.
Bibliography: 14 titles.