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. ^[1] 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 ${\displaystyle p}$ such that ${\displaystyle |y|\leq |x|}$ implies ${\displaystyle p(y)\leq p(x).}$ The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms. ^[1]

Properties

Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets. ^[1]

The strong dual of a locally convex vector lattice ${\displaystyle X}$ is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of ${\displaystyle X}$; moreover, if ${\displaystyle X}$ is a barreled space then the continuous dual space of ${\displaystyle X}$ is a band in the order dual of ${\displaystyle X}$ and the strong dual of ${\displaystyle X}$ is a complete locally convex TVS. ^[1]

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). ^[1]

If a locally convex vector lattice ${\displaystyle X}$ is semi-reflexive then it is order complete and ${\displaystyle X_{b}}$ (that is, ${\displaystyle \left(X,b\left(X,X^{\prime }\right)\right)}$) is a complete TVS; moreover, if in addition every positive linear functional on ${\displaystyle X}$ is continuous then ${\displaystyle X}$ is of ${\displaystyle X}$ is of minimal type, the order topology ${\displaystyle \tau _{\operatorname {O} }}$ on ${\displaystyle X}$ is equal to the Mackey topology ${\displaystyle \tau \left(X,X^{\prime }\right),}$ and ${\displaystyle \left(X,\tau _{\operatorname {O} }\right)}$ is reflexive. ^[1] 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). ^[1]

If a locally convex vector lattice ${\displaystyle X}$ 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. ^[1]

If ${\displaystyle X}$ is a separable metrizable locally convex ordered topological vector space whose positive cone ${\displaystyle C}$ is a complete and total subset of ${\displaystyle X,}$ then the set of quasi-interior points of ${\displaystyle C}$ is dense in ${\displaystyle C.}$ ^[1]

Theorem ^[1]  — Suppose that ${\displaystyle X}$ is an order complete locally convex vector lattice with topology ${\displaystyle \tau }$ and endow the bidual ${\displaystyle X^{\prime \prime }}$ of ${\displaystyle X}$ with its natural topology (that is, the topology of uniform convergence on equicontinuous subsets of ${\displaystyle X^{\prime }}$) and canonical order (under which it becomes an order complete locally convex vector lattice). The following are equivalent:

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

Corollary ^[1]  — Let ${\displaystyle X}$ be an order complete vector lattice with a regular order. The following are equivalent:

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

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

If ${\displaystyle (X,\tau )}$ is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces ${\displaystyle \left(X_{\alpha }\right)_{\alpha \in A}}$ and a family of ${\displaystyle A}$-indexed vector lattice embeddings ${\displaystyle f_{\alpha }:C_{\mathbb {R} }\left(K_{\alpha }\right)\to X}$ such that ${\displaystyle \tau }$ is the finest locally convex topology on ${\displaystyle X}$ making each ${\displaystyle f_{\alpha }}$ continuous. ^[2]

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

1. Schaefer & Wolff 1999, pp. 234–242.
2. Schaefer & Wolff 1999, pp. 242–250.

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