Jump to content

Order complete

From Wikipedia, the free encyclopedia

In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in (meaning contained in an interval, which is a set of the form for some ), the supremum ' and the infimum both exist and are elements of An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself,[1][2] in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum.[1]

Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.

Examples

[edit]

The order dual of a vector lattice is an order complete vector lattice under its canonical ordering.[1]

If is a locally convex topological vector lattice then the strong dual is an order complete locally convex topological vector lattice under its canonical order.[3]

Every reflexive locally convex topological vector lattice is order complete and a complete TVS.[3]

Properties

[edit]

If is an order complete vector lattice then for any subset is the ordered direct sum of the band generated by and of the band of all elements that are disjoint from [1] For any subset of the band generated by is [1] If and are lattice disjoint then the band generated by contains and is lattice disjoint from the band generated by which contains [1]

See also

[edit]
  • Vector lattice – Partially ordered vector space, ordered as a lattice

References

[edit]
  1. ^ a b c d e f Schaefer & Wolff 1999, pp. 204–214.
  2. ^ Narici & Beckenstein 2011, pp. 139–153.
  3. ^ a b Schaefer & Wolff 1999, pp. 234–239.

Bibliography

[edit]
  • 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.