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 [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ |y| \leq |x| }[/math] implies [math]\displaystyle{ p(y) \leq p(x). }[/math] 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 [math]\displaystyle{ X }[/math] is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of [math]\displaystyle{ X }[/math]; moreover, if [math]\displaystyle{ X }[/math] is a barreled space then the continuous dual space of [math]\displaystyle{ X }[/math] is a band in the order dual of [math]\displaystyle{ X }[/math] and the strong dual of [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] is semi-reflexive then it is order complete and [math]\displaystyle{ X_b }[/math] (that is, [math]\displaystyle{ \left( X, b\left(X, X^{\prime}\right) \right) }[/math]) is a complete TVS; moreover, if in addition every positive linear functional on [math]\displaystyle{ X }[/math] is continuous then [math]\displaystyle{ X }[/math] is of [math]\displaystyle{ X }[/math] is of minimal type, the order topology [math]\displaystyle{ \tau_{\operatorname{O}} }[/math] on [math]\displaystyle{ X }[/math] is equal to the Mackey topology [math]\displaystyle{ \tau\left(X, X^{\prime}\right), }[/math] and [math]\displaystyle{ \left(X, \tau_{\operatorname{O}}\right) }[/math] is reflexive. 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 (that is, the strong dual of the strong dual).

If a locally convex vector lattice [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] is a separable metrizable locally convex ordered topological vector space whose positive cone [math]\displaystyle{ C }[/math] is a complete and total subset of [math]\displaystyle{ X, }[/math] then the set of quasi-interior points of [math]\displaystyle{ C }[/math] is dense in [math]\displaystyle{ C. }[/math]

Theorem — Suppose that [math]\displaystyle{ X }[/math] is an order complete locally convex vector lattice with topology [math]\displaystyle{ \tau }[/math] and endow the bidual [math]\displaystyle{ X^{\prime\prime} }[/math] of [math]\displaystyle{ X }[/math] with its natural topology (that is, the topology of uniform convergence on equicontinuous subsets of [math]\displaystyle{ X^{\prime} }[/math]) and canonical order (under which it becomes an order complete locally convex vector lattice). The following are equivalent:

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

Corollary — Let [math]\displaystyle{ X }[/math] be an order complete vector lattice with a regular order. The following are equivalent:

1. [math]\displaystyle{ X }[/math] is of minimal type.
2. For every majorized and direct subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ X, }[/math] the section filter of [math]\displaystyle{ S }[/math] converges in [math]\displaystyle{ X }[/math] when [math]\displaystyle{ X }[/math] is endowed with the order topology.
3. Every order convergent filter in [math]\displaystyle{ X }[/math] converges in [math]\displaystyle{ X }[/math] when [math]\displaystyle{ X }[/math] is endowed with the order topology.

Moreover, if [math]\displaystyle{ X }[/math] is of minimal type then the order topology on [math]\displaystyle{ X }[/math] is the finest locally convex topology on [math]\displaystyle{ X }[/math] for which every order convergent filter converges.

If [math]\displaystyle{ (X, \tau) }[/math] is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces [math]\displaystyle{ \left(X_{\alpha}\right)_{\alpha \in A} }[/math] and a family of [math]\displaystyle{ A }[/math]-indexed vector lattice embeddings [math]\displaystyle{ f_{\alpha} : C_{\R}\left(K_{\alpha}\right) \to X }[/math] such that [math]\displaystyle{ \tau }[/math] is the finest locally convex topology on [math]\displaystyle{ X }[/math] making each [math]\displaystyle{ f_{\alpha} }[/math] continuous.

Examples

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

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Fréchet lattice

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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.