Geodesic
Locally minimal topological groups and their embeddings into products of $o$-bounded groups
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 811-815.

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It is proven that an infinite-dimensional Banach space (considered as an Abelian topological group) is not topologically isomorphic to a subgroup of a product of $\sigma $-compact (or more generally, $o$-bounded) topological groups. This answers a question of M. Tkachenko.
Classification : 22A05, 22E15, 54H11
Keywords: $\omega$-bounded group; $\sigma$-bounded group; $o$-bounded group; Weil complete group; locally minimal group; Lie group 
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Banakh, T. Locally minimal topological groups and their embeddings into products of $o$-bounded groups. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 811-815. http://geodesic.mathdoc.fr/item/CMUC_2000__41_4_a13/