ordered topological vector space

{{Main|Normal cone (functional analysis)}}

If C is a cone in a TVS X then C is normal if $\backslash mathcal\{U\}\; =\; \backslash left[\; \backslash mathcal\{U\}\; \backslash right]\_\{C\}$, where $\backslash mathcal\{U\}$ is the neighborhood filter at the origin, $\backslash left[\; \backslash mathcal\{U\}\; \backslash right]\_\{C\}\; =\; \backslash left\backslash \{\; \backslash left[\; U\; \backslash right]\; :\; U\; \backslash in\; \backslash mathcal\{U\}\; \backslash right\backslash \}$, and $[U]\_\{C\}\; :=\; \backslash left(U\; +\; C\backslash right)\; \backslash cap\; \backslash left(U\; -\; C\backslash right)$ is the C-saturated hull of a subset U of X.{{sfn | Schaefer | Wolff | 1999 | pp=215–222}}

If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:{{sfn | Schaefer | Wolff | 1999 | pp=215–222}}

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

and if X is a vector space over the reals then also:{{sfn | Schaefer | Wolff | 1999 | pp=215–222}}

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

If the topology on X is locally convex then the closure of a normal cone is a normal cone.{{sfn | Schaefer | Wolff | 1999 | pp=215–222}}

If C is a normal cone in X and B is a bounded subset of X then $\backslash left[\; B\; \backslash right]\_\{C\}$ is bounded; in particular, every interval $[a,\; b]$ is bounded.{{sfn | Schaefer | Wolff | 1999 | pp=215–222}}

If X is Hausdorff then every normal cone in X is a proper cone.{{sfn | Schaefer | Wolff | 1999 | pp=215–222}}

• 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.{{sfn | Schaefer | Wolff | 1999 | pp=222–225}}
• Let X be an ordered vector space over the reals with positive cone C. Then the following are equivalent:{{sfn | Schaefer | Wolff | 1999 | pp=222–225}}

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

• {{annotated link|Generalised metric}}
• {{annotated link|Order topology (functional analysis)}}
• {{annotated link|Ordered field}}
• {{annotated link|Ordered group}}
• {{annotated link|Ordered ring}}
• {{annotated link|Ordered vector space}}
• {{annotated link|Partially ordered space}}
• {{annotated link|Riesz space}}
• {{annotated link|Topological vector lattice}}

{{reflist}}

• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}

{{Functional analysis}}

{{Ordered topological vector spaces}}

{{Topological vector spaces}}

{{Order theory}}

Category:Functional analysis

Category:Order theory

Category:Topological vector spaces