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, if A and B are subsets of X, we say that A absorbs B if there exists a positive real number r such that

$\forall \alpha \in \mathbb {F} :|\alpha |\geq r\Rightarrow B\subset \alpha A.$

a set S is called absorbing if for all $x\in X$ there exists a positive real number r such that

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

with

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

Properties

If X is a vector space over real or complex numbers. S is convex subset of X. Then the following two conditions are equivalent.

• S is absorbing.

• $\forall x\in X,\exists \alpha \in \mathbb {R} ^{+},\,x\in \alpha S.$

The intersection of finite but nonempty family of absorbing sets is absorbing.

Every absorbing set contains 0.

Examples

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

See also

• Algebraic interior
• Bounded set (topological vector space)

References

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