---
tag: 0T0EJDGHMB
kind: definition
name: Normal convergence
msc: [40A30]
defines: 0EJVD9K0K1
uses: [0MRE6XRE5V, 0NX5NWAFC9]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_convergence
    locator: Definition
---

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ be bounded functions for $n \ge 0$.
The [[series of functions|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$ converges normally on $S$ if
$$\sum_{n=0}^{\infty} \sup_{x \in S} |f_n(x)| < \infty.$$

## Remarks

By the [[Weierstrass M-test|0NX5NWAFC9]], a normally convergent series converges absolutely and uniformly on $S$.
The converse fails. For constant functions $f_n = (-1)^n / (n + 1)$, the series converges uniformly but not normally.
René Baire introduced the concept in 1908.

## Explanations

### L1 human

Normal convergence means that the sizes of the functions, measured by their suprema, form a convergent series.
It is a simple condition that is stronger than uniform convergence.