The Sup Metric and Maximum Coordinate Error

The Largest Coordinate Difference

When two vectors have many coordinates, their discrepancy can be measured in different ways. One useful choice is to ask only for the largest absolute error in any single coordinate. This measure is called the sup metric. It is useful when each coordinate matters separately and the goal is to control the worst coordinate error, rather than the total error across all coordinates.

Fix \(n\in\mathbb{N}\). We work on \(\mathbb{R}^n\), whose elements are vectors \(x=(x_1,\dots,x_n)\). For \(x,y\in\mathbb{R}^n\), define their sup distance by taking the maximum of the \(n\) absolute coordinate differences. Because there are finitely many coordinates, this maximum exists.

The definition of an open set requires the whole radius-\(r\) neighborhood of \(x\) to be contained in \(U\), not merely that its points satisfy the distance inequality. The central computation for this metric is that the largest coordinate difference controls all the others: each individual error is at most \(d_\infty(x,y)\), and at least one coordinate attains that maximum.

Why the Sup Distance Is a Metric

Proof. Let \(x,y,z\in\mathbb{R}^n\). Each number \(|x_i-y_i|\) is nonnegative, so their maximum \(d_\infty(x,y)\) is nonnegative. If \(x=y\), then \(x_i=y_i\) for each \(i\), so every coordinate difference is \(0\) and \(d_\infty(x,y)=0\). Conversely, if \(d_\infty(x,y)=0\), then \(0\leq |x_i-y_i|\leq d_\infty(x,y)=0\) for every \(i\). Hence \(|x_i-y_i|=0\), so \(x_i=y_i\) for every coordinate and \(x=y\). Symmetry follows from \(|x_i-y_i|=|y_i-x_i|\) for each \(i\), which gives \(d_\infty(x,y)=d_\infty(y,x)\).

For the triangle inequality, the ordinary absolute-value triangle inequality gives, for every coordinate \(i\), \[ |x_i-y_i|\leq |x_i-z_i|+|z_i-y_i|. \] By the definition of a maximum, \(|x_i-z_i|\leq d_\infty(x,z)\) and \(|z_i-y_i|\leq d_\infty(z,y)\). Therefore, for every \(i\), \[ |x_i-y_i|\leq d_\infty(x,z)+d_\infty(z,y). \] Taking the maximum over all \(i\) on the left preserves this upper bound, so \(d_\infty(x,y)\leq d_\infty(x,z)+d_\infty(z,y)\). All four metric properties hold, proving that \(d_\infty\) is a metric. \(\square\)

The triangle-inequality proof relies on one important feature: the coordinate index set is finite, so \(d_\infty\) is a finite maximum. In particular, the proof compares every coordinate against the same two distances, then takes the maximum only after establishing the bound for each coordinate.

Distance Computation and Coordinate Bounds

Proof. First suppose \(d_\infty(x,y)\leq\delta\). Each coordinate difference is at most the maximum, so \(|x_i-y_i|\leq d_\infty(x,y)\leq\delta\) for every \(i\). Conversely, suppose \(|x_i-y_i|\leq\delta\) for every \(i\). The maximum of these finitely many values is then at most \(\delta\), giving \(d_\infty(x,y)\leq\delta\). For the strict version, if \(d_\infty(x,y)<\delta\), each coordinate difference is at most that maximum and hence is strictly less than \(\delta\). In the other direction, if every coordinate difference is strictly less than \(\delta\), their finite set has a largest value, and that largest value is one of the coordinate differences. It too is strictly less than \(\delta\), so \(d_\infty(x,y)<\delta\). The non-strict equivalence also covers \(\delta=0\): all coordinate differences must be zero, which means \(x=y\). \(\square\)

The distinction between strict and non-strict bounds matters in applications. A guarantee that every coordinate error is at most \(\delta\) gives a distance at most \(\delta\); a guarantee that every coordinate error is strictly less than \(\delta\) gives a distance strictly less than \(\delta\). The first statement does not imply the second: one coordinate could attain the bound exactly.

What the Maximum Measures

The sup metric answers a worst-coordinate question. If \(d_\infty(x,y)\leq\delta\), then no coordinate changes by more than \(\delta\). Conversely, if no coordinate changes by more than \(\delta\), the distance is at most \(\delta\). This is an exact equivalence, not an estimate that loses information. When the coordinates represent separate measurements, it provides a direct way to state a uniform error tolerance across all measurements.

For comparison, in \(\mathbb{R}^2\) the coordinate errors \(3\) and \(4\) have maximum \(4\), so the sup distance is \(4\). Adding those errors would give \(3+4=7\), a different quantity. That sum is not the definition of the sup metric. The maximum records the largest single-coordinate error; it does not accumulate errors across coordinates.

Common pitfall. A wrong calculation for vectors whose coordinate differences are \(3\) and \(4\) is to write \(d_\infty=3+4=7\). This adds the errors instead of taking their maximum. The correction is \(d_\infty=\max(3,4)=4\). A separate but related mistake is to replace “at most \(\delta\)” by “strictly less than \(\delta\).” In the first worked example, the largest error equals \(2\), so the distance is at most \(2\) but not strictly less than \(2\).

For quick reference, the rule for computing a sup distance is:

1. Subtract corresponding coordinates.
2. Take the absolute value of each difference.
3. Choose the largest of those finitely many nonnegative values.

The same maximum then gives the coordinatewise error guarantee: each coordinate error is bounded by the distance, and the distance itself is bounded by any common upper bound for all coordinate errors.