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 there exists a real number r such that
with
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 .
Examples
- In a semi normed vector space the unit ball is absorbing.
Properties
- The finite intersection of absorbing sets is absorbing
See also
References
- Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 4.
- Schaefer, Helmut H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 11. ISBN 0-387-98726-6.
This article is issued from Wikipedia - version of the 4/17/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.