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.
The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm.
The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.
[1] 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
is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of
is a band in the order dual of
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
is semi-reflexive then it is order complete and
) is a complete TVS; moreover, if in addition every positive linear functional on
is of minimal type, the order topology
is equal to the Mackey topology
[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
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.
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 an order complete locally convex vector lattice with topology
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).
is of minimal type then the order topology on
is the finest locally convex topology on
is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces
-indexed vector lattice embeddings
is the finest locally convex topology on