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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. LCVLs are important in the theory of topological vector lattices.

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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 {\displaystyle p} such that | y | ≤ | x | {\displaystyle |y|\leq |x|} implies p ( y ) ≤ p ( x ) . {\displaystyle p(y)\leq p(x).} The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.2

Properties

Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets.3

The strong dual of a locally convex vector lattice X {\displaystyle 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 {\displaystyle X} ; moreover, if X {\displaystyle X} is a barreled space then the continuous dual space of X {\displaystyle X} is a band in the order dual of X {\displaystyle X} and the strong dual of X {\displaystyle X} is a complete locally convex TVS.4

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).5

If a locally convex vector lattice X {\displaystyle X} is semi-reflexive then it is order complete and X b {\displaystyle X_{b}} (that is, ( X , b ( X , X ′ ) ) {\displaystyle \left(X,b\left(X,X^{\prime }\right)\right)} ) is a complete TVS; moreover, if in addition every positive linear functional on X {\displaystyle X} is continuous then X {\displaystyle X} is of X {\displaystyle X} is of minimal type, the order topology τ O {\displaystyle \tau _{\operatorname {O} }} on X {\displaystyle X} is equal to the Mackey topology τ ( X , X ′ ) , {\displaystyle \tau \left(X,X^{\prime }\right),} and ( X , τ O ) {\displaystyle \left(X,\tau _{\operatorname {O} }\right)} is reflexive.6 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).7

If a locally convex vector lattice X {\displaystyle 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.8

If X {\displaystyle X} is a separable metrizable locally convex ordered topological vector space whose positive cone C {\displaystyle C} is a complete and total subset of X , {\displaystyle X,} then the set of quasi-interior points of C {\displaystyle C} is dense in C . {\displaystyle C.} 9

Theorem10—Suppose that X {\displaystyle X} is an order complete locally convex vector lattice with topology τ {\displaystyle \tau } and endow the bidual X ′ ′ {\displaystyle X^{\prime \prime }} of X {\displaystyle X} with its natural topology (that is, the topology of uniform convergence on equicontinuous subsets of X ′ {\displaystyle 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 → X ′ ′ {\displaystyle X\to X^{\prime \prime }} induces an isomorphism of X {\displaystyle X} with an order complete sublattice of X ′ ′ . {\displaystyle X^{\prime \prime }.}
  2. For every majorized and directed subset S {\displaystyle S} of X , {\displaystyle X,} the section filter of S {\displaystyle S} converges in ( X , τ ) {\displaystyle (X,\tau )} (in which case it necessarily converges to sup S {\displaystyle \sup S} ).
  3. Every order convergent filter in X {\displaystyle X} converges in ( X , τ ) {\displaystyle (X,\tau )} (in which case it necessarily converges to its order limit).

Corollary11—Let X {\displaystyle X} be an order complete vector lattice with a regular order. The following are equivalent:

  1. X {\displaystyle X} is of minimal type.
  2. For every majorized and direct subset S {\displaystyle S} of X , {\displaystyle X,} the section filter of S {\displaystyle S} converges in X {\displaystyle X} when X {\displaystyle X} is endowed with the order topology.
  3. Every order convergent filter in X {\displaystyle X} converges in X {\displaystyle X} when X {\displaystyle X} is endowed with the order topology.

Moreover, if X {\displaystyle X} is of minimal type then the order topology on X {\displaystyle X} is the finest locally convex topology on X {\displaystyle X} for which every order convergent filter converges.

If ( X , τ ) {\displaystyle (X,\tau )} is a locally convex vector lattice that is bornological and sequentially complete, then there exists a family of compact spaces ( X α ) α ∈ A {\displaystyle \left(X_{\alpha }\right)_{\alpha \in A}} and a family of A {\displaystyle A} -indexed vector lattice embeddings f α : C R ( K α ) → X {\displaystyle f_{\alpha }:C_{\mathbb {R} }\left(K_{\alpha }\right)\to X} such that τ {\displaystyle \tau } is the finest locally convex topology on X {\displaystyle X} making each f α {\displaystyle f_{\alpha }} continuous.12

Examples

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

See also

Bibliography

  • 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.

References

  1. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  2. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  3. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  4. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  5. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  6. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  7. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  8. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  9. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  10. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  11. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

  12. Schaefer & Wolff 1999, pp. 242–250. - 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. https://search.worldcat.org/oclc/840278135 ↩