Locally convex vector lattice

In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices.

Properties

Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets.

If a locally convex vector lattice is barreled then its strong dual space is complete (this is not necessarily true if the space is merely a locally convex barreled space but not a locally convex vector lattice).

If a locally convex vector lattice X is semi-reflexive then it is order complete and $X_{b}$ (i.e. $\left(X,b\left(X,X^{\prime }\right)\right)$) is a complete TVS. Every reflexive locally convex vector lattice is order complete and a complete locally convex TVS whose strong dual is a barreled reflexive locally convex TVS that can be identified under the canonical evaluation map with the strong bidual (i.e. the strong dual of the strong dual).

Examples

Every Banach lattice, normed lattice, and Fréchet lattice is a locally convex vector lattice.

See also

• Banach lattice
• Fréchet lattice
• Normed lattice
• Vector lattice

References

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)