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 (TVS) topology on

[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

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] 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.

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

with its order topology (which makes it into a normable space).

is a regularly ordered vector space over the reals and if

is any subset of the positive cone

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.

is the finest locally convex topology on

is a normal cone; it is also the same as the Mackey topology induced on

is an ordered Fréchet lattice over the real numbers then

is a regularly ordered vector lattice then the ordered topology is the finest locally convex TVS topology on

into a locally convex vector lattice.

with the order topology is a barreled space and every band decomposition of

[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

will be an ordered vector space and

will denote the order topology on

is the quotient of the order topology on

[4] 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).