Pretty much what the title says.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have a topological vector space $(X,\tau)$ and $U\subseteq X$ is topologically bounded. Is it possible for there to be some $x\in X$ such that $cx\in U$ for arbitrarily large $c$? I'm thinking of a real vector space here.

If we try to prove this BWOC, suppose $U$ is topologically bounded but contains such an $x$. Right now I've gotten to the point where every neighborhood of the origin has to contain $cx$ for arbitrarily large $c$. This seems silly but... I see no contradiction.

How about if we add $(X,\tau)$ is topologically bounded? Or if that's not sufficient, what else should we add?

Sorry about the poor format. I don't see how to make the forum recognize my tex.

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# Can a topologically bounded set in a tvs contain a ray?

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