Order topology (functional analysis)
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 topology on for which every order interval is bounded, where an order interval in is a set of the form where and belong to
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.
Definitions
The family of all locally convex topologies on for which every order interval is bounded is non-empty and the order topology is the upper bound of this family.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
A neighborhood of the origin in the order topology is necessarily an absorbing set because for all
For every let and endow with its order topology.
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, then with its order topology is the inductive limit of .
The lattice structure can compensate in part for any lack of an order unit:
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
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.
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
Properties
Throughout, will be an ordered vector space and will denote the order topology on- The dual of is the order bound dual of
- If separates points in then is a bornological locally convex TVS.
- Each positive linear operator between two ordered vector spaces is continuous for the respective order topologies.
- Each order unit of an ordered TVS is interior to the positive cone for the order topology.
- If the order of an ordered vector space is a regular order and if each positive sequence of type in is order summable, then endowed with its order topology is a barreled space.
- If the order of an ordered vector space is a regular order and if for all and holds, then the positive cone of is a normal cone in when is endowed with the order topology. In particular, the continuous dual space of with the order topology will be the order dual +.
- If is an Archimedean ordered vector space over the real numbers having an order unit and let denote the order topology on Then is an ordered TVS that is normable, is the finest locally convex TVS topology on such that the positive cone is normal, and the following are equivalent:
- is complete.
- Each positive sequence of type in is order summable.
- In particular, if is an Archimedean ordered vector space having an order unit then the order is a regular order and
- If is a Banach space and an ordered vector space with an order unit then 's topological is identical to the order topology if and only if the positive cone of is a normal cone in
- A vector lattice homomorphism from into is a topological homomorphism when and are given their respective order topologies.