---
tag: 4R6SQD60FN
kind: concept
name: Bornological space
msc: [46A08, 46A17]
specializes: [B6GY2KJMWW]
sources:
  - url: https://en.wikipedia.org/wiki/Bornological_space
same_as:
  wikidata: Q894008
  wikipedia: Bornological space
---

## Summary

A bornological space is a [[locally convex space|B6GY2KJMWW]] whose topology is determined by its [[bounded sets|3BAJ8ZTVPX]].
In such a space, every convex balanced set that absorbs all bounded sets is a neighbourhood of $0$.
The main consequence concerns a linear map from a bornological space to a locally convex space.
Such a map is continuous as soon as it maps bounded sets to bounded sets.
Every metrizable locally convex space, in particular every normed space, is bornological.

## Explanations

### L1 human

For linear maps between normed spaces, continuity is the same as sending bounded sets to bounded sets.
Bornological spaces are the locally convex spaces where this convenient test still works.
They are the spaces whose topology can be read off from the knowledge of which sets are bounded.
A Banach space with its weak topology is a standard example that is not bornological when it is infinite-dimensional.