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

Suppose that is a vector space over the field of real or complex numbers.
Notation: If is a set and is a scalar then let.

One set absorbing another

Definition: If and are subsets of, we say that absorbs if it satisfies any of the following equivalent conditions:

  1. there exists a real such that for any scalar satisfying ;
  2. there exists a real such that for any scalar satisfying ;
and if is balanced then we can add to this list:

  1. there exists a scalar such that ;
  2. there exists a scalar such that.

Absorbing set

Definition: A set is called absorbing or absorbent in a vector space over a field if it satisfies any of the following equivalent conditions:

  1. for all, absorbs ;
  2. for all there exists a real such that for any scalar satisfying ;
  3. for all there exists a real such that for any scalar satisfying ;
  4. the algebraic interior of contains the origin ;
  5. for all, is a neighborhood of 0 in when is given its unique Hausdorff TVS topology;
and if is balanced then we can add to this list:

  1. for all, there exists a scalar such that ;
  2. for all, there exists a real such that ;
and if is convex then we can add to this list:

  1. for all, there exists a real such that .

Examples

Conditions for one set to absorb another

Sufficient conditions to be absorbing

Properties

Every absorbing set contains the origin.
If is an absorbing disk in a vector space then there exists an absorbing disk in such that.
Let be a vector space over real or complex numbers.
If is convex subset of then the following two conditions are equivalent:

  1. is absorbing in ;
  2. for all, there exists some such that.