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, $$\left( X, b\left(X, X^{\prime}\right) \right)$$) 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 $$\tau_{\operatorname{O}}$$ on $$X$$ is equal to the Mackey topology $$\tau\left(X, X^{\prime}\right),$$ and $$\left(X, \tau_{\operatorname{O}}\right)$$ 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 $$C.$$

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

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