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 1 Definition 2 Examples 3 Properties 4 See also 5 References

[edit] 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

[edit] Examples

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

[edit] Properties

• The finite intersection of absorbing sets is absorbing

[edit] See also

• Bounded set (topological vector space)

[edit] References

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 4. 
• Schaefer, Helmuth H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 11. ISBN 0-387-98726-6.