Are the finite dimensional Banach spaces precisely those ones in which subsets are totally bounded iff they're bounded?
2026-03-31 22:25:31.1774995931
Banach Spaces: Totally Bounded vs. Bounded
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I'm assuming a normed space $X$ based on your comments.
If $X$ is an infinite-dimensional normed space, then there exists a sequence $\{ x_{n}\}_{n=1}^{\infty}$ of unit vectors such that $\|x_{m}-x_{n}\| \ge 1/2$ for all $n \ne m$. That's a consequence of the Riesz lemma. I think that answers your question.