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

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

with

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

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 ${\displaystyle T\subseteq rS}$.

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

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 4.
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 11. ISBN 0-387-98726-6.