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 $xin\; X$ there exists a real number "r" such that:$forall\; alpha\; in\; mathbb\{F\}\; :\; vert\; alpha\; vert\; ge\; r\; Rightarrow\; x\; in\; alpha\; S$with:$alpha\; S\; :=\; \{\; alpha\; s\; mid\; s\; in\; S\}$

Examples

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

Properties

*The finite intersection of absorbing sets is absorbing

ee also

*Bounded set (topological vector space)

References

*
*