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Locally convex vector lattice

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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.

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.[1]

Properties

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 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.[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 X is semi-reflexive then it is order complete and (i.e. ) is a complete TVS.[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 (i.e. the strong dual of the strong dual).[1]

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.[1]

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.[2]

Theorem:[2] Suppose that X is an order complete locally convex vector lattice with topology 𝜏 and endow the bidual of X with its natural topology (i.e. 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:

  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 (X, 𝜏) (in which case it necessarily converges to ).
  3. Every order convergent filter in X converges in (X, 𝜏) (in which case it necessarily converges to its order limit).

Examples

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

See also

References

  1. ^ a b c d e f g h Schaefer & Wolff 1999, pp. 234–242.
  2. ^ a b Schaefer 1999, pp. 234–242.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)