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:

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:

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$$\sqrt{n}\Vert \mathbf{x}{\Vert }_{\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 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$$\Vert \mathbf{x}{\Vert }_2$.

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

Examples¶

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

Prerequisites¶

• Real Coordinate Space
• Norm and Normed Vector Space
• Metric and Metric Space
• Euclidean Norm

Related Concepts¶

• $p$-norm and $p$-metric
• Taxicab Metric
• Uniform Norm

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