Is the following statement true? The closed unit ball of a normed space is compact

No. The unit ball of ℓ² Riesz: compactness of the ball holds exactly in finite dimensions.

Counterexample

Is this true?

The closed unit ball of a normed space is compact

No — it is false.

The counterexample

The unit ball of $\ell^{2}$

Why it fails

Riesz: compactness of the ball holds exactly in finite dimensions.

Keep practicing

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.