Order complete

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

If ${\displaystyle X}$ is a locally convex topological vector lattice then the strong dual ${\displaystyle X_{b}^{\prime }}$ 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]

If ${\displaystyle X}$ is an order complete vector lattice then for any subset ${\displaystyle S\subseteq X,}$ ${\displaystyle X}$ is the ordered direct sum of the band generated by ${\displaystyle A}$ and of the band ${\displaystyle A^{\perp }}$ of all elements that are disjoint from ${\displaystyle A.}$[1] For any subset ${\displaystyle A}$ of ${\displaystyle X,}$ the band generated by ${\displaystyle A}$ is ${\displaystyle A^{\perp \perp }.}$[1] If ${\displaystyle x}$ and ${\displaystyle y}$ are lattice disjoint then the band generated by ${\displaystyle \{x\},}$ contains ${\displaystyle y}$ and is lattice disjoint from the band generated by ${\displaystyle \{y\},}$ which contains ${\displaystyle x.}$[1]

• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

• 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.