---
tag: 044D4AQV8V
kind: definition
name: Product metric
aliases: [max-product-metric]
msc: ["54E35"]
defines: 0AYV6VGKF5
uses: [063W5JMY85, 0C81YBVTWH]
sources:
  - url: https://en.wikipedia.org/wiki/Product_metric
    locator: lead section
---

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].
The product metric on the [[Cartesian product|0C81YBVTWH]] $X \times Y$ is the function $d$ with
$$d\bigl((x_1, y_1), (x_2, y_2)\bigr) = \max\bigl(d_X(x_1, x_2), \, d_Y(y_1, y_2)\bigr)$$
for all $x_1, x_2 \in X$ and $y_1, y_2 \in Y$.

## Remarks

The function $d$ is a metric on $X \times Y$.
Other common product metrics are $d_X + d_Y$ and $\sqrt{d_X^2 + d_Y^2}$. Each of them is between $d$ and $2d$, so all of them are [[equivalent|0TWQ5PRKWS]] and give the [[product topology|0Z57EC6YX3]].
For $X = Y = \mathbb{R}$ with the usual metric, $d$ is the [[sup metric|0VTAXQEWQD]] on $\mathbb{R}^2$.
The same formula with a maximum over $n$ factors gives a product metric on a product of $n$ metric spaces.

## Explanations

### L1 human

Two pairs are close in the product metric when their first parts are close and their second parts are close.
The maximum is the worse of the two distances.