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.

Lattice semi-norms

The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm p such that $\left|y\right|\leq \left|x\right|$ implies $p(y)\leq p(x)$. The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.

Properties

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

The strong dual of a locally convex vector lattice X is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of X; moreover, if X is a barreled space then the continuous dual space of X is a band in the order dual of X and the strong dual of X is a complete locally convex TVS.

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).

If a locally convex vector lattice X is an infrabarreled TVS then it can be identified under the evaluation map with a topological vector sublattice of its strong bidual, which is an order complete locally convex vector lattice under its canonical order.

If X is a separable metrizable locally convex ordered topological vector space whose positive cone C is a complete and total subset of X, then the set of quasi-interior points of C is dense in C.

Theorem: Suppose that X is an order complete locally convex vector lattice with topology 𝜏 and endow the bidual $X^{\prime \prime }$ of X with its natural topology (i.e. the topology of uniform convergence on equicontinuous subsets of $X^{\prime }$) and canonical order (under which it becomes an order complete locally convex vector lattice). The following are equivalent:

1. The evaluation map $X\to X^{\prime \prime }$ induces an isomorphism of X with an order complete sublattice of $X^{\prime \prime }$.
2. For every majorized and directed subset S of X, the section filter of S converges in (X, 𝜏) (in which case it necessarily converges to $\sup S$).
3. Every order convergent filter in X converges in (X, 𝜏) (in which case it necessarily converges to its order limit).

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)