This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.Find sources: "Order topology" functional analysis – news · newspapers · books · scholar · JSTOR (June 2020)

In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space is the finest locally convex topological vector space (TVS) topology on for which every order interval is bounded, where an order interval in is a set of the form where and belong to [1]

The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of rather than from some topology that starts out having. This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology.[2]

Definitions

The family of all locally convex topologies on for which every order interval is bounded is non-empty (since it contains the coarsest possible topology on ) and the order topology is the upper bound of this family.[1]

A subset of is a neighborhood of the origin in the order topology if and only if it is convex and absorbs every order interval in [1] A neighborhood of the origin in the order topology is necessarily an absorbing set because for all [1]

For every let and endow with its order topology (which makes it into a normable space). The set of all 's is directed under inclusion and if then the natural inclusion of into is continuous. If is a regularly ordered vector space over the reals and if is any subset of the positive cone of that is cofinal in (e.g. could be ), then with its order topology is the inductive limit of (where the bonding maps are the natural inclusions).[3]

The lattice structure can compensate in part for any lack of an order unit:

Theorem[3] — Let be a vector lattice with a regular order and let denote its positive cone. Then the order topology on is the finest locally convex topology on for which is a normal cone; it is also the same as the Mackey topology induced on with respect to the duality

In particular, if is an ordered Fréchet lattice over the real numbers then is the ordered topology on if and only if the positive cone of is a normal cone in [3]

If is a regularly ordered vector lattice then the ordered topology is the finest locally convex TVS topology on making into a locally convex vector lattice. If in addition is order complete then with the order topology is a barreled space and every band decomposition of is a topological direct sum for this topology.[3] In particular, if the order of a vector lattice is regular then the order topology is generated by the family of all lattice seminorms on [3]

Properties

Throughout, will be an ordered vector space and will denote the order topology on

  1. is complete.
  2. Each positive sequence of type in is order summable.

Relation to subspaces, quotients, and products

If is a solid vector subspace of a vector lattice then the order topology of is the quotient of the order topology on [4]

Examples

The order topology of a finite product of ordered vector spaces (this product having its canonical order) is identical to the product topology of the topological product of the constituent ordered vector spaces (when each is given its order topology).[3]

See also

References

  1. ^ a b c d Schaefer & Wolff 1999, pp. 204–214.
  2. ^ Schaefer & Wolff 1999, p. 204.
  3. ^ a b c d e f g h i j k l m n o Schaefer & Wolff 1999, pp. 230–234.
  4. ^ a b Schaefer & Wolff 1999, pp. 250–257.

Bibliography