Just to be sure: In an infinite dimensional Banach space the open unit ball cannot be totally bounded, right? The context is that I need this in order to find a lack in here...
2026-03-31 18:57:34.1774983454
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Banach Space: Open Unit Ball Totally Bounded?
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You are right: it is not totally bounded. Riesz's lemma directly leads to an infinite uniformly separated subset of unit ball, as the Wikipedia article shows.
If the open unit ball were totally bounded, then so would its closure, which is the closed unit ball. The closed unit ball, in turn, is complete (as a closed subset of a Banach space). Hence, the closed unit ball would be compact, which it can be shown it is not, given that the space is infinite-dimensional.