Maximum Norm¶

Definition¶

The maximum norm of $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)$ in an $n$ -dimensional real coordinate space $\mathbb{R}^n$ with fixed Cartesian coordinate system is:

\|\boldsymbol{x}\|_\infty=\max_{1\leq i\leq n}|x_i|.

Correspondingly, the Chebyshev distance between $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)$ and $\boldsymbol{y}=(y_1,y_2,\ldots,y_n)$ is calculated as:

d_\infty(\boldsymbol{x},\boldsymbol{y})=\max_{1\leq i\leq n}|x_i-y_i|.

This is a norm because it is zero only at the zero vector, scales with absolute scalar size, and obeys the triangle inequality coordinate by coordinate. The induced distance is a metric distance. Comparison with the Euclidean Norm gives $\|\boldsymbol{x}\|_\infty\le\|\boldsymbol{x}\|_2\le\sqrt n\,\|\boldsymbol{x}\|_\infty$ $n ∥ x ∥_{\infty}$ .

Interpretation¶

The maximum norm measures a vector by its largest coordinate in absolute value; the other coordinates do not matter as long as they are smaller. It is the limiting case of the $p$ -norm as $p\to+\infty$ $+ \infty$ .

The square with corners at plus and minus 1, the unit ball of the maximum norm, containing the Euclidean unit circle, with the point (0.6, -1) on its edge.

The square is the closed unit ball $\{\boldsymbol{x}\colon\|\boldsymbol{x}\|_\infty\le1\}$ $1}$ : a point lies in it exactly when both coordinates lie in $[-1,1]$ . The marked point $\boldsymbol{x}=(0.6,-1)$ lies on the edge because its largest coordinate in absolute value is $|-1|=1$ $1$ ; the coordinate $0.6$ does not change the norm. The dashed Euclidean unit circle lies inside the square, which is the inequality $\|\boldsymbol{x}\|_\infty\le\|\boldsymbol{x}\|_2$ $∥ x ∥_{2}$ .

With $\|\boldsymbol x\|_\infty=\max(|x_1|,|x_2|)$ the unit circle is a square. Drag $\boldsymbol x$ and compare the three norms.

The Chebyshev distance $d_\infty(\boldsymbol{x},\boldsymbol{y})$ is small if and only if every coordinate of $\boldsymbol{x}$ is close to the corresponding coordinate of $\boldsymbol{y}$ . This makes it convenient in proofs: $d_\infty(\boldsymbol{x}_k,\boldsymbol{x})\to0$ says exactly that each coordinate sequence converges, and since the maximum norm is equivalent to the Euclidean norm, convergence in $\mathbb{R}^n$ is the same as convergence of each coordinate.

For functions, taking the largest absolute value becomes the uniform norm $\|f\|_\infty=\sup_x|f(x)|$ $sup_{x} ∣ f (x) ∣$ : the values $f(x)$ play the role of the coordinates, and the maximum becomes a supremum because there may be infinitely many of them.

A common slip is to take the maximum of the signed coordinates. The absolute values come first: $\|(-3,2)\|_\infty=3$ $3$ , not $2$ .

Key Results¶

The closed unit ball in the maximum norm is an axis-aligned hypercube

\overline{B}_{d_\infty}(\boldsymbol{0},1)=[-1,1]^n.

Examples¶

$\|(-2,1)\|_\infty=\max\{2,1\}=2$ $2$ .
$d_\infty((1,-1),\,(3,0))=2$ $2$ .

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 4

Aliases: max norm, infinity norm on R^n Keywords: Chebyshev distance, maximum metric Symbols: infinity, real numbers, square root

Last updated: 20 October 2026 03:02