It is proved that every real metrizable locally convex space which is not nuclear contains a closed additive subgroup K such that the quotient group G = (span K)/K admits a non-trivial continuous positive definite function, but no non-trivial continuous character. Consequently, G cannot satisfy any form of the Bochner theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm170-3-5,
author = {Robert Stegli\'nski},
title = {Quotient groups of non-nuclear spaces for which the Bochner theorem fails completely},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {283-295},
zbl = {1080.43006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm170-3-5}
}
Robert Stegliński. Quotient groups of non-nuclear spaces for which the Bochner theorem fails completely. Studia Mathematica, Tome 166 (2005) pp. 283-295. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm170-3-5/