Exercise 1068Topology › Metric and normed spacesL1Added 3 Nov 2026

Equivalent norms

#equivalent norms#topology#finite dimension

Statement

Show that in finite-dimensional spaces, any two norms are equivalent.

Hint

Work in a basis; the closed unit ball of one norm is compact for the other.

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Solution

Setup. Let $\|\cdot\|_1,\|\cdot\|_2$ be two norms on $\mathbb{R}^n$ . By compactness of the unit sphere of $\|\cdot\|_1$ (for $\|\cdot\|_2$ ), the continuous function $\|\cdot\|_1/\|\cdot\|_2$ has a positive minimum $m$ and finite maximum $M$ .

Consequence. $m\|x\|_2\le\|x\|_1\le M\|x\|_2$ .

Conclusion. In finite dimension, all norms are equivalent.

Note. Fails in infinite dimension (e.g. $\|\cdot\|_\infty$ vs $\|\cdot\|_1$ on $C([0,1])$ ).

Equivalent norms

Statement

Solution

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