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Barrelled set

From Wikipedia, the free encyclopedia

In functional analysis, a subset of a topological vector space (TVS) is called a barrel or a barrelled set if it is closed convex balanced and absorbing.

Barrelled sets play an important role in the definitions of several classes of topological vector spaces, such as barrelled spaces.

Definitions

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Let be a topological vector space (TVS). A subset of is called a barrel if it is closed convex balanced and absorbing in A subset of is called bornivorous[1] and a bornivore if it absorbs every bounded subset of Every bornivorous subset of is necessarily an absorbing subset of

Let be a subset of a topological vector space If is a balanced absorbing subset of and if there exists a sequence of balanced absorbing subsets of such that for all then is called a suprabarrel[2] in where moreover, is said to be a(n):

  • bornivorous suprabarrel if in addition every is a closed and bornivorous subset of for every [2]
  • ultrabarrel if in addition every is a closed subset of for every [2]
  • bornivorous ultrabarrel if in addition every is a closed and bornivorous subset of for every [2]

In this case, is called a defining sequence for [2]

Properties

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Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.

Examples

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See also

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References

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  1. ^ Narici & Beckenstein 2011, pp. 441–457.
  2. ^ a b c d e Khaleelulla 1982, p. 65.

Bibliography

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  • Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • H.H. Schaefer (1970). Topological Vector Spaces. GTM. Vol. 3. Springer-Verlag. ISBN 0-387-05380-8.
  • Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. Vol. 936. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.