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 $p$ 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 $X$ 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.[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 $X$ is semi-reflexive then it is order complete and $X_{b}$ (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 $X$ is continuous then $X$ is of $X$ is of minimal type, the order topology ${\displaystyle \tau _{\operatorname {O} ))$ on $X$ 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 $X$ 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 $X$ 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 ${\displaystyle C.}$[1]

Theorem[1] — Suppose that $X$ is an order complete locally convex vector lattice with topology $\tau$ and endow the bidual ${\displaystyle X^{\prime \prime ))$ of $X$ 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 $X$ with an order complete sublattice of ${\displaystyle X^{\prime \prime }.}$
2. For every majorized and directed subset $S$ of $X,$ the section filter of $S$ converges in $(X,\tau )$ (in which case it necessarily converges to ${\displaystyle \sup S}$).
3. Every order convergent filter in $X$ converges in $(X,\tau )$ (in which case it necessarily converges to its order limit).

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

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

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

If $(X,\tau )$ 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 $A$-indexed vector lattice embeddings ${\displaystyle f_{\alpha }:C_{\mathbb {R} }\left(K_{\alpha }\right)\to X}$ such that $\tau$ is the finest locally convex topology on $X$ making each ${\displaystyle f_{\alpha ))$ continuous.[2]

## Examples

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