Definition of Absorbing Set in Topology Vector Space

In summary, an absorbing set in a topological vector space is defined as a set A that satisfies X = \bigcup_{n\in \mathbb N} nA, meaning that any element of X can eventually be absorbed by a scaled version of A. While some sources may define absorbing sets as being able to be arbitrarily expanded by any scalar, this definition is still widely accepted. Additionally, it can be proven that every neighborhood of zero in a topological vector space is absorbing.
  • #1
AxiomOfChoice
533
1
Is this a legitimate definition for an "absorbing set" in a topological vector space?

A set [itex]A\subset X[/itex] is absorbing if [itex]X = \bigcup_{n\in \mathbb N} nA[/itex].

This is the definition the way it was presented to us in my functional analysis class, but I'm looking at other sources, and it seems everyone talks about absorbing being something that can be arbitrarily expanded (by *any* scalar) to include the whole space. It seems that expansion just by natural numbers is too restrictive.
 
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  • #2


My professor also claims that EVERY neighborhood of zero in a topological vector space is absorbing. This wasn't proved, and I can't think of why it would be the case.
 
  • #3


Because if [tex]X = \bigcup_{n\in \mathbb N} nA[/tex], then any element of X eventually belongs (is absorbed) by a scaled version nA of A. A little thinking also shows that any open neighborhood is absorbing.
 

1. What is an absorbing set in topology vector space?

An absorbing set in topology vector space is a subset of the vector space that can absorb any element of the vector space by scalar multiplication. This means that for any vector in the vector space, there exists a scalar such that when the vector is multiplied by this scalar, it falls within the absorbing set.

2. What is the significance of an absorbing set in topology vector space?

An absorbing set is important in topology vector space because it helps define the topology of the space. It allows for the definition of open sets and convergence of sequences, which are fundamental concepts in topology. Absorbing sets also help in proving the continuity of vector space operations and functions.

3. How is an absorbing set different from a bounded set?

An absorbing set is different from a bounded set in that it is not limited by a specific distance or size. A bounded set has a finite radius or diameter, whereas an absorbing set can stretch to cover the entire vector space. Absorbing sets are also used in infinite-dimensional vector spaces, where the concept of boundedness may not apply.

4. Can an absorbing set be empty?

Yes, an absorbing set can be empty. An absorbing set is defined as a subset of a vector space, and any set can be empty. An empty absorbing set would mean that there is no scalar that can absorb any element of the vector space, making it a trivial case.

5. How are absorbing sets related to convex sets?

Absorbing sets and convex sets are closely related in topology vector space. In fact, every absorbing set is a convex set, but not every convex set is an absorbing set. This means that an absorbing set is a special type of convex set that has the additional property of absorbing all the elements in the vector space.

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