The main result: the dual of separable Banach space X contains a total subspace which is not norming over any infinite-dimensional subspace of X if and only if X has a nonquasireflexive quotient space with a strictly singular quotient mapping.
@article{bwmeta1.element.bwnjournal-article-smv105i1p37bwm,
author = {M. Ostrovski\u\i },
title = {Total subspaces in dual Banach spaces which are not norming over any infinite-dimensional subspace},
journal = {Studia Mathematica},
volume = {104},
year = {1993},
pages = {37-49},
zbl = {0810.46016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv105i1p37bwm}
}
Ostrovskiĭ, M. Total subspaces in dual Banach spaces which are not norming over any infinite-dimensional subspace. Studia Mathematica, Tome 104 (1993) pp. 37-49. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv105i1p37bwm/
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