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Is this true?

The closed unit ball of a normed space is compact

No — it is false.

The counterexample

The unit ball of

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

The kind of mistake this is

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this
topologycompactness

Others that fail the same way

From Topology › Compactness and Tychonoff

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