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.

Contents

1 Definition

Definition

Given a vector space X over a field F a set S is called absorbing if for all x there exists a real number r such that

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

with

$\alpha S := \{ \alpha x \mid x \in S\}$