Normal cone (functional analysis)

If $C$ is a cone in a TVS $X$ then for any subset $S\; \backslash subseteq\; X$ let $[S]\_C\; :=\; \backslash left(S\; +\; C\backslash right)\; \backslash cap\; \backslash left(S\; -\; C\backslash right)$ be the $C$-saturated hull of $S\; \backslash subseteq\; X$ and for any collection $\backslash mathcal\{S\}$ of subsets of $X$ let $\backslash left[\; \backslash mathcal\{S\}\; \backslash right]\_C\; :=\; \backslash left\backslash \{\; \backslash left[\; S\; \backslash right]\_C\; :\; S\; \backslash in\; \backslash mathcal\{S\}\; \backslash right\backslash \}.$

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.{{sfn|Schaefer|Wolff|1999|pp=215–222}}

If $\backslash mathcal\{T\}$ is a collection of subsets of $X$ and if $\backslash mathcal\{F\}$ is a subset of $\backslash mathcal\{T\}$ then $\backslash mathcal\{F\}$ is a fundamental subfamily of $\backslash mathcal\{T\}$ if every $T\; \backslash in\; \backslash mathcal\{T\}$ is contained as a subset of some element of $\backslash mathcal\{F\}.$

If $\backslash mathcal\{G\}$ is a family of subsets of a TVS $X$ then a cone $C$ in $X$ is called a $\backslash mathcal\{G\}$-cone if $\backslash left\backslash \{\; \backslash overline\{\backslash left[\; G\; \backslash right]\_C\}\; :\; G\; \backslash in\; \backslash mathcal\{G\}\; \backslash right\backslash \}$ is a fundamental subfamily of $\backslash mathcal\{G\}$ and $C$ is a strict $\backslash mathcal\{G\}$-cone if $\backslash left\backslash \{\; \backslash left[\; G\; \backslash right]\_C\; :\; G\; \backslash in\; \backslash mathcal\{G\}\; \backslash right\backslash \}$ is a fundamental subfamily of $\backslash mathcal\{G\}.${{sfn|Schaefer|Wolff|1999|pp=215–222}}

Let $\backslash mathcal\{B\}$ denote the family of all bounded subsets of $X.$

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 $\backslash mathcal\{G\}$ in $X$ such that $B\; \backslash in\; \backslash mathcal\{G\}$ implies $\backslash left[\; B\; \backslash cap\; C\; \backslash right]\_C\; \backslash subseteq\; B.$

and if $X$ is a vector space over the reals then we may add to this list:{{sfn|Schaefer|Wolff|1999|pp=215–222}}

4. There exists a neighborhood base at the origin consisting of convex, balanced, $C$-saturated sets.
5. 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 $p\; \backslash in\; \backslash mathcal\{P\}.$

and if $X$ is a locally convex space and if the dual cone of $C$ is denoted by $X^\{\backslash prime\}$ then we may add to this list:{{sfn|Schaefer|Wolff|1999|pp=215–222}}

6. For any equicontinuous subset $S\; \backslash subseteq\; X^\{\backslash prime\},$ there exists an equicontiuous $B\; \backslash subseteq\; C^\{\backslash prime\}$ such that $S\; \backslash subseteq\; B\; -\; B.$
7. The topology of $X$ is the topology of uniform convergence on the equicontinuous subsets of $C^\{\backslash prime\}.$

and if $X$ is an infrabarreled locally convex space and if $\backslash mathcal\{B\}^\{\backslash prime\}$ is the family of all strongly bounded subsets of $X^\{\backslash prime\}$ then we may add to this list:{{sfn|Schaefer|Wolff|1999|pp=215–222}}

8. The topology of $X$ is the topology of uniform convergence on strongly bounded subsets of $C^\{\backslash prime\}.$
9. $C^\{\backslash prime\}$ is a $\backslash mathcal\{B\}^\{\backslash prime\}$-cone in $X^\{\backslash prime\}.$
   • this means that the family $\backslash left\backslash \{\; \backslash overline\{\backslash left[\; B^\{\backslash prime\}\; \backslash right]\_C\}\; :\; B^\{\backslash prime\}\; \backslash in\; \backslash mathcal\{B\}^\{\backslash prime\}\; \backslash right\backslash \}$ is a fundamental subfamily of $\backslash mathcal\{B\}^\{\backslash prime\}.$

10. $C^\{\backslash prime\}$ is a strict $\backslash mathcal\{B\}^\{\backslash prime\}$-cone in $X^\{\backslash prime\}.$
    • this means that the family $\backslash left\backslash \{\; \backslash left[\; B^\{\backslash prime\}\; \backslash right]\_C\; :\; B^\{\backslash prime\}\; \backslash in\; \backslash mathcal\{B\}^\{\backslash prime\}\; \backslash right\backslash \}$ is a fundamental subfamily of $\backslash mathcal\{B\}^\{\backslash prime\}.$

and if $X$ is an ordered locally convex TVS over the reals whose positive cone is $C,$ then we may add to this list:

11. there exists a Hausdorff locally compact topological space $S$ such that $X$ is isomorphic (as an ordered TVS) with a subspace of $R(S),$ where $R(S)$ is the space of all real-valued continuous functions on $X$ under the topology of compact convergence.{{sfn|Schaefer|Wolff|1999|pp=222-225}}

If $X$ is a locally convex TVS, $C$ is a cone in $X$ with dual cone $C^\{\backslash prime\}\; \backslash subseteq\; X^\{\backslash prime\},$ and $\backslash mathcal\{G\}$ is a saturated family of weakly bounded subsets of $X^\{\backslash prime\},$ then{{sfn|Schaefer|Wolff|1999|pp=215–222}}

1. if $C^\{\backslash prime\}$ is a $\backslash mathcal\{G\}$-cone then $C$ is a normal cone for the $\backslash mathcal\{G\}$-topology on $X$;
2. if $C$ is a normal cone for a $\backslash mathcal\{G\}$-topology on $X$ consistent with $\backslash left\backslash langle\; X,\; X^\{\backslash prime\}\backslash right\backslash rangle$ then $C^\{\backslash prime\}$ is a strict $\backslash mathcal\{G\}$-cone in $X^\{\backslash prime\}.$

If $X$ is a Banach space, $C$ is a closed cone in $X,$, and $\backslash mathcal\{B\}^\{\backslash prime\}$ is the family of all bounded subsets of $X^\{\backslash prime\}\_b$ then the dual cone $C^\{\backslash prime\}$ is normal in $X^\{\backslash prime\}\_b$ if and only if $C$ is a strict $\backslash mathcal\{B\}$-cone.{{sfn|Schaefer|Wolff|1999|pp=215–222}}

If $X$ is a Banach space and $C$ is a cone in $X$ then the following are equivalent:{{sfn|Schaefer|Wolff|1999|pp=215–222}}

1. $C$ is a $\backslash mathcal\{B\}$-cone in $X$;
2. $X\; =\; \backslash overline\{C\}\; -\; \backslash overline\{C\}$;
3. $\backslash overline\{C\}$ is a strict $\backslash mathcal\{B\}$-cone in $X.$

Suppose $L$ is an ordered topological vector space. That is, $L$ is a topological vector space, and we define $x\; \backslash geq\; y$ whenever $x\; -\; y$ lies in the cone $L\_+$. The following statements are equivalent:{{Cite book |last=Aliprantis |first=Charalambos D. |url=https://www.worldcat.org/oclc/87808043 |title=Cones and duality |date=2007 |publisher=American Mathematical Society |others=Rabee Tourky |isbn=978-0-8218-4146-4 |location=Providence, R.I. |oclc=87808043}}

1. The cone $L\_+$ is normal;
2. The normed space $L$ admits an equivalent monotone norm;
3. There exists a constant $c\; >\; 0$ such that $a\; \backslash leq\; x\; \backslash leq\; b$ implies $\backslash lVert\; x\; \backslash rVert\; \backslash leq\; c\; \backslash max\backslash \{\backslash lVert\; a\; \backslash rVert,\; \backslash lVert\; b\; \backslash rVert\backslash \}$;
4. The full hull $[U]\; =\; (U\; +\; L\_+)\; \backslash cap\; (U\; -\; L\_+)$ of the closed unit ball $U$ of $L$ is norm bounded;
5. There is a constant $c\; >\; 0$ such that $0\; \backslash leq\; x\; \backslash leq\; y$ implies $\backslash lVert\; x\; \backslash rVert\; \backslash leq\; c\; \backslash lVert\; y\; \backslash rVert$.

• If $X$ is a Hausdorff TVS then every normal cone in $X$ is a proper cone.{{sfn|Schaefer|Wolff|1999|pp=215–222}}
• If $X$ is a normable space and if $C$ is a normal cone in $X$ then $X^\{\backslash prime\}\; =\; C^\{\backslash prime\}\; -\; C^\{\backslash prime\}.${{sfn|Schaefer|Wolff|1999|pp=215–222}}
• Suppose that the positive cone of an ordered locally convex TVS $X$ is weakly normal in $X$ and that $Y$ is an ordered locally convex TVS with positive cone $D.$ If $Y\; =\; D\; -\; D$ then $H\; -\; H$ is dense in $L\_s(X;\; Y)$ where $H$ is the canonical positive cone of $L(X;\; Y)$ and $L\_\{s\}(X;\; Y)$ is the space $L(X;\; Y)$ with the topology of simple convergence.{{sfn|Schaefer|Wolff|1999|pp=225–229}}
• If $\backslash mathcal\{G\}$ is a family of bounded subsets of $X,$ then there are apparently no simple conditions guaranteeing that $H$ is a $\backslash mathcal\{T\}$-cone in $L\_\{\backslash mathcal\{G\}\}(X;\; Y),$ even for the most common types of families $\backslash mathcal\{T\}$ of bounded subsets of $L\_\{\backslash mathcal\{G\}\}(X;\; Y)$ (except for very special cases).{{sfn|Schaefer|Wolff|1999|pp=225–229}}

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}}

Suppose that $\backslash left\backslash \{\; X\_\{\backslash alpha\}\; :\; \backslash alpha\; \backslash in\; A\; \backslash right\backslash \}$ is a family of locally convex TVSs and that $C\_\backslash alpha$ is a cone in $X\_\{\backslash alpha\}.$

If $X\; :=\; \backslash bigoplus\_\{\backslash alpha\}\; X\_\{\backslash alpha\}$ is the locally convex direct sum then the cone $C\; :=\; \backslash bigoplus\_\{\backslash alpha\}\; C\_\backslash alpha$ is a normal cone in $X$ if and only if each $X\_\{\backslash alpha\}$ is normal in $X\_\{\backslash alpha\}.${{sfn|Schaefer|Wolff|1999|pp=215–222}}

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

If $C$ is a cone in a locally convex TVS $X$ and if $C^\{\backslash prime\}$ is the dual cone of $C,$ then $X^\{\backslash prime\}\; =\; C^\{\backslash prime\}\; -\; C^\{\backslash prime\}$ if and only if $C$ is weakly normal.{{sfn|Schaefer|Wolff|1999|pp=215–222}}

Every normal cone in a locally convex TVS is weakly normal.{{sfn|Schaefer|Wolff|1999|pp=215–222}}

In a normed space, a cone is normal if and only if it is weakly normal.{{sfn|Schaefer|Wolff|1999|pp=215–222}}

If $X$ and $Y$ are ordered locally convex TVSs and if $\backslash mathcal\{G\}$ is a family of bounded subsets of $X,$ then if the positive cone of $X$ is a $\backslash mathcal\{G\}$-cone in $X$ and if the positive cone of $Y$ is a normal cone in $Y$ then the positive cone of $L\_\{\backslash mathcal\{G\}\}(X;\; Y)$ is a normal cone for the $\backslash mathcal\{G\}$-topology on $L(X;\; Y).${{sfn|Schaefer|Wolff|1999|pp=225-229}}

• {{annotated link|Cone-saturated}}
• {{annotated link|Topological vector lattice}}
• {{annotated link|Vector lattice}}

{{reflist}}

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

{{Functional analysis}}

{{Ordered topological vector spaces}}

Category:Functional analysis