locally convex vector lattice

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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 |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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

Properties

Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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).{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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).{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

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.{{sfn|Schaefer|Wolff|1999|pp=234–242}}

{{math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=234–242}}|note=|style=|math_statement=

Suppose that X is an order complete locally convex vector lattice with topology \tau and endow the bidual X^{\prime\prime} of X with its natural topology (that is, the topology of uniform convergence on equicontinuous subsets of X^{\prime}) and canonical order (under which it becomes an order complete locally convex vector lattice). The following are equivalent:

  1. The evaluation map X \to X^{\prime\prime} induces an isomorphism of X with an order complete sublattice of 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 \sup S).
  3. Every order convergent filter in X converges in (X, \tau) (in which case it necessarily converges to its order limit).

}}

{{math theorem|name=Corollary{{sfn|Schaefer|Wolff|1999|pp=234–242}}|note=|style=|math_statement=

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 \left(X_{\alpha}\right)_{\alpha \in A} 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.{{sfn|Schaefer|Wolff|1999|pp=242–250}}

Examples

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

See also

  • {{annotated link|Banach lattice}}
  • {{annotated link|Fréchet lattice}}
  • {{annotated link|Normed lattice}}
  • {{annotated link|Vector lattice}}

References

{{reflist|group=note}}

{{reflist}}

Bibliography

  • {{Narici Beckenstein Topological Vector Spaces|edition=2}}
  • {{Schaefer Wolff Topological Vector Spaces|edition=2}}

{{Functional analysis}}

{{Ordered topological vector spaces}}

{{Order theory}}

Category:Functional analysis