Absorbing set

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 Swith: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

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