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.
The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm such that implies The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.
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 is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of ; moreover, if is a barreled space then the continuous dual space of is a band in the order dual of and the strong dual of 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 is semi-reflexive then it is order complete and (that is, ) is a complete TVS; moreover, if in addition every positive linear functional on is continuous then is of is of minimal type, the order topology on is equal to the Mackey topology and 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 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 is a separable metrizable locally convex ordered topological vector space whose positive cone is a complete and total subset of then the set of quasi-interior points of is dense in 
Theorem — Suppose that is an order complete locally convex vector lattice with topology and endow the bidual of with its natural topology (that is, the topology of uniform convergence on equicontinuous subsets of ) and canonical order (under which it becomes an order complete locally convex vector lattice). The following are equivalent:
Moreover, if is of minimal type then the order topology on is the finest locally convex topology on for which every order convergent filter converges.
If is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces and a family of -indexed vector lattice embeddings such that is the finest locally convex topology on making each continuous.
Every Banach lattice, normed lattice, and Fréchet lattice is a locally convex vector lattice.