Exercise 515Topology › Metric and normed spacesL1Added 3 Nov 2026

Classical metrics on $\mathbb{R}^n$

#Euclidean#Manhattan#sup norm

Statement

Define the Euclidean, Manhattan, and sup metrics on $\mathbb{R}^n$ and compare them.

Hint

$\|x\|_2\le\|x\|_1\le n\|x\|_2$ and $\|x\|_\infty\le\|x\|_2\le\sqrt n\|x\|_\infty$ .

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Metrics.
- Euclidean: $d_2(x,y)=\sqrt{\sum(x_i-y_i)^2}$ .
- Manhattan: $d_1(x,y)=\sum|x_i-y_i|$ .
- Sup: $d_\infty(x,y)=\max_i|x_i-y_i|$ .

Comparison. For every $x\in\mathbb{R}^n$ :
$\|x\|_\infty\le\|x\|_2\le\|x\|_1\le n\|x\|_\infty.$

Conclusion. These metrics define the same topology (they are equivalent).