---
tag: 04GM4NYXJB
kind: definition
name: Mesh of a partition
aliases: [norm-of-a-partition]
msc: [26A42]
defines: 093SAXPJ9P
uses: [069P20KYQT]
sources:
  - url: https://en.wikipedia.org/wiki/Partition_of_an_interval
    locator: Norm of a partition
---

## Definition

Let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
The mesh of $P$ is
$$\| P \| = \max_{1 \le i \le n} (x_i - x_{i-1}),$$
the length of the longest subinterval of $P$.

## Remarks

Refining a partition does not increase its mesh.
The [[Riemann integral|0QM4TXT3DR]] is a limit of Riemann sums as the mesh tends to $0$.
For the partition of $[0, 1]$ into $n$ equal pieces, the mesh is $1/n$.

## Explanations

### L1 human

The mesh measures how fine a partition is.
A small mesh means that all the pieces are short.