Order convergence

In mathematics, specifically in order theory and functional analysis, a filter in an order complete vector lattice X is order convergent if it contains an order bounded subset (i.e. is contained in an interval of the form [a,b] = { xX : axb }) 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 X).[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

A net in a vector lattice X is said to decrease to if implies and in X. A net in a vector lattice X is said to order-converge to if there is a net in X that decreases to 0 and satisfies for all .[2]

Order continuity

A linear T : X Y between vector lattices is said to be order continuous if whenever is a net in X that order-converges to x0 in X, then the net order-converges to T(x0) in Y. T is said to be sequentially order continuous if whenever is a sequence in X that order-converges to x0 in X, then the sequence order-converges to T(x0) in Y.[2]

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

See also

References

  1. Schaefer & Wolff 1999, pp. 234–242.
  2. Khaleelulla 1982, p. 8.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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