Absorbing set

In mathematics, a subset of a real or complex vector space is called absorbing if there exists for every vector in a positive real number such that an element of is for every real or complex with.

The name comes from the fact that the above definition is equivalent to the following is: For all in there so that an element of is for all. ( It obviously suffices to replace the above. ) That is, the amount may be "inflated" by scalar multiplication so that it contains any vector and not lose again by further inflation, ie him "absorb" can.

This second formulation seems more natural at first glance. The former definition is preferred because it can be transmitted naturally to the definition of a limited quantity of a topological module ( namely, an amount that is absorbed by every neighborhood of zero ). Because of the possible existence of zero divisors and the possible non- existence of bounded neighborhoods of zero, in this case is not a useful definition for the purposes of the second formulation.

Simple consequences

In a topological vector space (eg in a normed space ) any neighborhood of zero is absorbing, as is a vector in, then, that for sufficiently large.