---
tag: NM1BCK3ZX9
kind: concept
name: Absorbing set
msc: [46A03, 52A07]
sources:
  - url: https://en.wikipedia.org/wiki/Absorbing_set
same_as:
  wikidata: Q332775
  wikipedia: Absorbing set
---

## Summary

A subset of a real or complex vector space is absorbing if every vector lies in a large enough multiple of it.
Every neighbourhood of $0$ in a topological vector space is absorbing.
The [[Minkowski functional|PAX9SHDHDW]] of an absorbing set is finite everywhere, so absorbing sets are the starting point for turning convex sets into [[seminorms|FP5GVCN8GV]].

## Explanations

### L1 human

An absorbing set reaches out a little in every direction from the origin.
Blow it up enough, and it swallows any given vector.
The open unit disc in $\R^2$ is absorbing: the point $(10, 0)$ lies in $11$ times the disc.
A line through the origin in $\R^2$ is not absorbing: no multiple of it contains a point off the line.