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 $x\in X$ there exists a real number r such that

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

with

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

Examples

• In a semi normed vector space the unit ball is absorbing.

Properties

• The finite intersection of absorbing sets is absorbing

See also

• Algebraic interior
• Bounded set (topological vector space)

References

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