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 $, y, \leq , x,$ 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 $X_b$ (that is, $\left( X, b\left(X, X^\right) \right)$) is a complete TVS;
moreover, if in addition every positive linear functional on $X$ is continuous then $X$ is of $X$ is of minimal type, the order topology $\tau_$ on $X$ is equal to the Mackey topology $\tau\left(X, X^\right),$ and $\left(X, \tau_\right)$ 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 $X$ 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 $C.$

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

Examples

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

See also

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References

Bibliography

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{{Order theory

Functional analysis