---
tag: ZE6DXEJB8H
kind: concept
name: Barrelled space
msc: [46A08]
specializes: [B6GY2KJMWW]
sources:
  - url: https://en.wikipedia.org/wiki/Barrelled_space
same_as:
  wikidata: Q2090232
  wikipedia: Barrelled space
---

## Summary

A barrel is a closed, convex, balanced and absorbing set.
A barrelled space is a [[locally convex space|B6GY2KJMWW]] in which every barrel is a neighbourhood of $0$.
Barrelled spaces are the spaces where the uniform boundedness principle (Banach–Steinhaus theorem) holds in its usual form.
Banach spaces, [[Fréchet spaces|3GAA614673]] and [[LF-spaces|7THJSH9SFP]] are barrelled.

## Explanations

### L1 human

A barrel is a set that looks like a closed unit ball: it is closed, convex, symmetric, and big enough in every direction.
In a barrelled space every such set is automatically a genuine neighbourhood of $0$.
This rules out sets that are "thin" in infinitely many directions at once.
The reason one cares is that pointwise bounded families of continuous linear maps are then equicontinuous.
In a Banach space this is the classical uniform boundedness principle, proved with the Baire category theorem.