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

Normal cone

If C is a cone in a TVS X then C is normal if ${\displaystyle {\mathcal {U}}=\left_{C}}$, where ${\displaystyle {\mathcal {U}}}$ is the neighborhood filter at the origin, ${\displaystyle \left_{C}=\left\{\left:U\in {\mathcal {U}}\right\}}$, and ${\displaystyle _{C}:=\left(U+C\right)\cap \left(U-C\right)}$ 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 ${\displaystyle {\mathcal {F}}}$ in X, if ${\displaystyle \lim {\mathcal {F}}=0}$ then ${\displaystyle \lim \left_{C}=0}$.
3. There exists a neighborhood base ${\displaystyle {\mathcal {B}}}$ in X such that ${\displaystyle B\in {\mathcal {B}}}$ implies ${\displaystyle \left_{C}\subseteq B}$.

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 ${\displaystyle {\mathcal {P}}}$ of semi-norms on X such that ${\displaystyle p(x)\leq p(x+y)}$ for all ${\displaystyle x,y\in C}$ and ${\displaystyle p\in {\mathcal {P}}}$.

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 ${\displaystyle \left_{C}}$ is bounded; in particular, every interval ${\displaystyle }$ 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 ${\displaystyle X^{+}}$ 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

• 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

1. ^ Schaefer & Wolff 1999, pp. 222–225.
2. ^ 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.

• Functional analysis
• Order theory
• Topological vector spaces