In mathematics, specifically infunctional analysis andorder theory, anordered topological vector space, also called anordered TVS, is atopological vector space (TVS)X that has apartial order ≤ making it into anordered vector space whose positive cone${\displaystyle C:=\left\{x\in X:x\ge 0\right\}}$ is a closed subset ofX.^[1] Ordered TVSes have important applications inspectral theory.

IfC is a cone in a TVSX thenC isnormal if${\displaystyle \mathcal{U}={\left[\mathcal{U}\right]}_C}$, where${\displaystyle \mathcal{U}}$ is the neighborhood filter at the origin,${\displaystyle {\left[\mathcal{U}\right]}_C=\left\{\left[U\right]:U\in \mathcal{U}\right\}}$, and${\displaystyle [U{]}_C:=\left(U+C\right)\cap \left(U-C\right)}$ is theC-saturated hull of a subsetU ofX.^[2]

IfC is a cone in a TVSX (over the real or complex numbers), then the following are equivalent:^[2]

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

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

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

IfC is a normal cone inX andB is a bounded subset ofX then${\displaystyle {\left[B\right]}_C}$ is bounded; in particular, every interval${\displaystyle [a,b]}$ is bounded.^[2] IfX is Hausdorff then every normal cone inX is a proper cone.^[2]

• LetX be anordered vector space over the reals that is finite-dimensional. Then the order ofX is Archimedean if and only if the positive cone ofX is closed for the unique topology under whichX is a Hausdorff TVS.^[1]
• LetX be an ordered vector space over the reals with positive coneC. Then the following are equivalent:^[1]

1. the order ofX is regular.
2. C is sequentially closed for some Hausdorff locally convex TVS topology onX and${\displaystyle X^{+}}$ distinguishes points inX
3. the order ofX is Archimedean andC is normal for some Hausdorff locally convex TVS topology onX.

• 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
• 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

• 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

• Pages displaying short descriptions of redirect targets via Module:Annotated link