Locally convex vector lattice


In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice 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 implies.
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 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.
If a locally convex vector lattice X is semi-reflexive then it is order complete and 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 on X 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.
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.
Theorem: Suppose that X is an order complete locally convex vector lattice with topology ? and endow the bidual of X with its natural topology and canonical order. The following are equivalent:

  1. The evaluation map induces an isomorphism of X with an order complete sublattice of.
  2. For every majorized and directed subset S of X, the section filter of S converges in .
  3. Every order convergent filter in X converges in .
Corollary: 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 is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces and a family of A-indexed vector lattice embeddings such that ? is the finest locally convex topology on X making each continuous.

Examples

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