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

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The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm   such that   implies   The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.[1]

Properties

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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  ; moreover, if   is a barreled space then the continuous dual space of   is a band in the order dual of   and the strong 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   (that is,  ) is a complete TVS; moreover, if in addition every positive linear functional on   is continuous then   is of   is of minimal type, the order topology   on   is equal to the Mackey topology   and   is reflexive.[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.[1]

If   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 dense in  [1]

Theorem[1] — Suppose that   is an order complete locally convex vector lattice with topology   and endow the bidual   of   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). The following are equivalent:

  1. The evaluation map   induces an isomorphism of   with an order complete sublattice of  
  2. For every majorized and directed subset   of   the section filter of   converges in   (in which case it necessarily converges to  ).
  3. Every order convergent filter in   converges in   (in which case it necessarily converges to its order limit).

Corollary[1] — Let   be an order complete vector lattice with a regular order. The following are equivalent:

  1.   is of minimal type.
  2. For every majorized and direct subset   of   the section filter of   converges in   when   is endowed with the order topology.
  3. Every order convergent filter in   converges in   when   is endowed with the order topology.

Moreover, if   is of minimal type then the order topology on   is the finest locally convex topology on   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  -indexed vector lattice embeddings   such that   is the finest locally convex topology on   making each   continuous.[2]

Examples

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

See also

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References

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  1. ^ a b c d e f g h i j k Schaefer & Wolff 1999, pp. 234–242.
  2. ^ Schaefer & Wolff 1999, pp. 242–250.

Bibliography

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  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.