Girish Mahajan (Editor)

Absorbing set

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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.

Contents

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 ∈ X there exists a real number r such that

∀ α ∈ F : | α | ≥ r ⇒ x ∈ α S

with

α S := { α s ∣ s ∈ 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 T ⊆ r S .

Examples

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

    • Properties

  • The finite intersection of absorbing sets is absorbing

    • References

    Absorbing set Wikipedia
    (Text) CC BY-SA