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  C := { x ∈ X : x ≥ 0 } {\displaystyle C:=\left\\{x\in X:x\geq 0\right\\}} is a closed subset of X.[1] Ordered TVSes have important applications in spectral theory.

## Normal cone

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Main article: Normal cone (functional analysis)

If C is a cone in a TVS X then C is normal if  U = [ U ] C {\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C}} , where  U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin,  [ U ] C = { [ U ] : U ∈ U } {\displaystyle \left[{\mathcal {U}}\right]_{C}=\left\\{\left[U\right]:U\in {\mathcal {U}}\right\\}} , and  [ U ] C := ( U + C ) ∩ ( U − C ) {\displaystyle [U]_{C}:=\left(U+C\right)\cap \left(U-C\right)} is the C-saturated hull of a subset U of X.[2]

If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:[2]

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

and if X is a vector space over the reals then also:[2]

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

If the topology on X is locally convex then the closure of a normal cone is a normal cone.[2]

### Properties

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If C is a normal cone in X and B is a bounded subset of X then  [ B ] C {\displaystyle \left[B\right]_{C}} is bounded; in particular, every interval  [ a , b ] {\displaystyle [a,b]} is bounded.[2] If X is Hausdorff then every normal cone in X is a proper cone.[2]

## Properties

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* 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.[1]
* Let X be an ordered vector space over the reals with positive cone C. Then the following are equivalent:[1]

1. the order of X is regular.
2. C is sequentially closed for some Hausdorff locally convex TVS topology on X and  X + {\displaystyle X^{+}} 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

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* Generalised metric – Metric geometry
* Order topology (functional analysis) – Topology of an ordered vector space
* Ordered field – Algebraic object with an ordered structure
* Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
* Ordered ring – ring with a compatible total orderPages displaying wikidata descriptions as a fallback
* 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

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1. ^ a b c Schaefer & Wolff 1999, pp. 222–225.
2. ^ a b c d e f Schaefer & Wolff 1999, pp. 215–222.

* 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.

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