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$\omega$-bounded group; $\sigma$-bounded group; $o$-bounded group; Weil complete group; locally minimal group; Lie group
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.
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