Exercise 1066Topology › Metric and normed spacesL1Added 3 Nov 2026

Equivalent metrics

#equivalent metrics#same topology#norms

Statement

Show that the three classical metrics on $\mathbb{R}^n$ are equivalent: they define the same topology.

Hint

Two metrics are topologically equivalent iff each open ball for one contains an open ball for the other.

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Bounds. $\|x\|_\infty\le\|x\|_2\le\sqrt n\|x\|_\infty$ , and similar for $\|\cdot\|_1$ .

Consequence. $B_\infty(a,r/\sqrt n)\subseteq B_2(a,r)\subseteq B_\infty(a,r)$ ; similarly for $\|\cdot\|_1$ .

Conclusion. $d_1,d_2,d_\infty$ are topologically equivalent on $\mathbb{R}^n$ .