Order convergence

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In mathematics, specifically in order theory and functional analysis, a filter in an order complete vector lattice is order convergent if it contains an order bounded subset (that is, a subset contained in an interval of the form ) and if where is the set of all order bounded subsets of X, in which case this common value is called the order limit of in [1]

Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.

Definition

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A net   in a vector lattice   is said to decrease to   if   implies   and   in   A net   in a vector lattice   is said to order-converge to   if there is a net   in   that decreases to   and satisfies   for all  .[2]

Order continuity

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A linear map   between vector lattices is said to be order continuous if whenever   is a net in   that order-converges to   in   then the net   order-converges to   in     is said to be sequentially order continuous if whenever   is a sequence in   that order-converges to   in  then the sequence   order-converges to   in  [2]

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In an order complete vector lattice   whose order is regular,   is of minimal type if and only if every order convergent filter in   converges when   is endowed with the order topology.[1]

See also

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References

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  1. ^ a b Schaefer & Wolff 1999, pp. 234–242.
  2. ^ a b Khaleelulla 1982, p. 8.
  • 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.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.