Geodesic
Reflexivity and best approximations in Fr\'echet spaces
Izvestiya. Mathematics , Tome 17 (1981) no. 1, pp. 87-94

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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. 
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     author = {D. N. Zarnadze},
     title = {Reflexivity and best approximations in {Fr\'echet} spaces},
     journal = {Izvestiya. Mathematics },
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D. N. Zarnadze. Reflexivity and best approximations in Fr\'echet spaces. Izvestiya. Mathematics , Tome 17 (1981) no. 1, pp. 87-94. http://geodesic.mathdoc.fr/item/IM2_1981_17_1_a3/