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

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

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

  • Vector lattice – Partially ordered vector space, ordered as a lattice

References

    Bibliography

    • 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.
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