Locally convex vector lattice explained

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

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.

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} 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.