---
tag: NWMYW2WK9S
kind: concept
name: Montel space
msc: [46A11]
specializes: [ZE6DXEJB8H]
sources:
  - url: https://en.wikipedia.org/wiki/Montel_space
same_as:
  wikidata: Q1516604
  wikipedia: Montel space
---

## Summary

A Montel space is a [[barrelled space|ZE6DXEJB8H]] in which every closed [[bounded set|3BAJ8ZTVPX]] is compact.
This is a Heine–Borel property, which fails in every infinite-dimensional normed space.
The name comes from [[Montel's theorem|ZVFVBVJMCB]]: the space of holomorphic functions on an open subset of $\C$ is a Montel space.
Spaces of smooth functions and of test functions used in distribution theory are Montel spaces too, and Montel spaces are reflexive.

## Explanations

### L1 human

In $\R^n$, a closed and bounded set is compact: every sequence in it has a convergent subsequence.
In an infinite-dimensional Banach space this fails, already for the closed unit ball.
Montel spaces are infinite-dimensional spaces where it holds again.
For holomorphic functions, a family that is bounded on each compact set has a subsequence that converges uniformly on compact sets.
That is the content of Montel's theorem, and it is why these spaces carry Montel's name.