While reading a proof on the closure of the span of finite number vectors in a hilbert space with respect to the norm induced topology, I became stumped on a particular step of the proof using the Bolzano Weierstrass theorem.(adsbygoogle = window.adsbygoogle || []).push({});

For finite dimensional vector spaces, Bolzano Weierstrass states that:

"If ||an||<=M, then an contains a convergent subsequence (wrt whatever metric is defined on this vector space)"

Is it safe to assume that the above theorem extends to infinite dimensional vectors spaces (Hilbert spaces)?

Thanks!

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# Question about the Bolzano Weierstrass analogue in Hilbert spaces

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