# Absorbing set

In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space. Alternative terms are radial or absorbent set.

## Definition

Given a vector space X over the field F of real or complex numbers, a set S is called absorbing if for all ${\displaystyle x\in X}$ there exists a real number r such that

${\displaystyle \forall \alpha \in \mathbb {F} :\vert \alpha \vert \geq r\Rightarrow x\in \alpha S}$

with

${\displaystyle \alpha S:=\{\alpha s\mid s\in S\}}$

The notion of the set S being absorbing is different from the notion that S absorbs some other subset T of X since the latter means that there exists some real number r > 0 such that ${\displaystyle T\subseteq rS}$.