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.
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]
and if X is a vector space over the reals then also:[2]
If the topology on X is locally convex then the closure of a normal cone is a normal cone.[2]
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]
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