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

Normal cone[edit]

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

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

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

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

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

Properties[edit]

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

Properties[edit]

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]

the order of X is regular.
C is sequentially closed for some Hausdorff locally convex TVS topology on X and $X^{+}$ distinguishes points in X
the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.

References[edit]

^ ^a ^b ^c Schaefer & Wolff 1999, pp. 222–225.
^ ^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.

[FOOTNOTESchaeferWolff1999222–225-1] Schaefer & Wolff 1999, pp. 222–225.

[FOOTNOTESchaeferWolff1999215–222-2] ^ ^a ^b ^c ^d ^e ^f Schaefer & Wolff 1999, pp. 215–222.

[1]

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Ordered topological vector spaces
Basic concepts	Ordered vector space Partially ordered space Riesz space Order topology Order unit Positive linear operator Topological vector lattice Vector lattice
Types of orders/spaces	AL-space AM-space Archimedean Banach lattice Fréchet lattice Locally convex vector lattice Normed lattice Order bound dual Order dual Order complete Regularly ordered
Types of elements/subsets	Band Cone-saturated Lattice disjoint Dual/Polar cone Normal cone Order complete Order summable Order unit Quasi-interior point Solid set Weak order unit
Topologies/Convergence	Order convergence Order topology
Operators	Positive State
Main results	Freudenthal spectral

Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

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