Ordered topological vector space

In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤ making it into an ordered vector space whose positive cone [math]\displaystyle{ C := \left\{ x \in X : x \geq 0\right\} }[/math] is a closed subset of X. Ordered TVS have important applications in spectral theory.

Normal cone

If C is a cone in a TVS X then C is normal if [math]\displaystyle{ \mathcal{U} = \left[ \mathcal{U} \right]_{C} }[/math], where [math]\displaystyle{ \mathcal{U} }[/math] is the neighborhood filter at the origin, [math]\displaystyle{ \left[ \mathcal{U} \right]_{C} = \left\{ \left[ U \right] : U \in \mathcal{U} \right\} }[/math], and [math]\displaystyle{ [U]_{C} := \left(U + C\right) \cap \left(U - C\right) }[/math] is the C-saturated hull of a subset U of X.

If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:

1. C is a normal cone.
2. For every filter [math]\displaystyle{ \mathcal{F} }[/math] in X, if [math]\displaystyle{ \lim \mathcal{F} = 0 }[/math] then [math]\displaystyle{ \lim \left[ \mathcal{F} \right]_{C} = 0 }[/math].
3. There exists a neighborhood base [math]\displaystyle{ \mathcal{B} }[/math] in X such that [math]\displaystyle{ B \in \mathcal{B} }[/math] implies [math]\displaystyle{ \left[ B \cap C \right]_{C} \subseteq B }[/math].

and if X is a vector space over the reals then also:

1. There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
2. There exists a generating family [math]\displaystyle{ \mathcal{P} }[/math] of semi-norms on X such that [math]\displaystyle{ p(x) \leq p(x + y) }[/math] for all [math]\displaystyle{ x, y \in C }[/math] and [math]\displaystyle{ p \in \mathcal{P} }[/math].

If the topology on X is locally convex then the closure of a normal cone is a normal cone.

Properties

If C is a normal cone in X and B is a bounded subset of X then [math]\displaystyle{ \left[ B \right]_{C} }[/math] is bounded; in particular, every interval [math]\displaystyle{ [a, b] }[/math] is bounded. If X is Hausdorff then every normal cone in X is a proper cone.

Properties

• Let X be an ordered vector space over the reals that is finite-dimensional. Then the order of X is Archimedean if and only if the positive cone of X is closed for the unique topology under which X is a Hausdorff TVS.
• Let X be an ordered vector space over the reals with positive cone C. Then the following are equivalent:

1. the order of X is regular.
2. C is sequentially closed for some Hausdorff locally convex TVS topology on X and [math]\displaystyle{ X^{+} }[/math] distinguishes points in X
3. the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.

See also

• Generalised metric
• Order topology (functional analysis) – Topology of an ordered vector space
• Ordered field – Algebraic object with an ordered structure
• Ordered ring
• Ordered vector space – Vector space with a partial order
• Partially ordered space – Partially ordered topological space
• Riesz space – Partially ordered vector space, ordered as a lattice
• Topological vector lattice

References

• 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.