This is about an exercise in [Bass]:

Exercise 19.5. Prove that if is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in is not compact.

Proof. Choose an orthonormal basis , then . This means the sequence is not Cauchy hence has no convergent subsequence.

For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.

 

[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.