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.