Category: Real Analysis

[Short Notes] Non-compactness of the closed unit ball in an infinite-dimensional Banach space

This is about an exercise in [Bass]:

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

Proof. Choose an orthonormal basis \{x_i\}, then ||x_i-x_j||^2=||x_i||^2+||x_j||^2=2. 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.