View source for Normal cone (functional analysis)

In mathematics, specifically in [[Order theory|order theory]] and [[Functional analysis|functional analysis]], if <math>C</math> is a cone at the origin in a [[Topological vector space|topological vector space]] <math>X</math> such that <math>0 \in C</math> and if <math>\mathcal{U}</math> is the neighborhood filter at the origin, then <math>C</math> is called '''normal''' if <math>\mathcal{U} = \left[ \mathcal{U} \right]_C,</math> where <math>\left[ \mathcal{U} \right]_C := \left\{ [ U ]_C : U \in \mathcal{U} \right\}</math> and where for any subset <math>S \subseteq X,</math> <math>[S]_C := (S + C) \cap (S - C)</math> is the [[Cone-saturated|<math>C</math>-saturatation]] of <math>S.</math>{{sfn|Schaefer|Wolff|1999|pp=215–222}}

Normal cones play an important role in the theory of [[Ordered topological vector space|ordered topological vector space]]s and [[Topological vector lattice|topological vector lattice]]s.

== Characterizations ==

If <math>C</math> is a cone in a TVS <math>X</math> then for any subset <math>S \subseteq X</math> let <math>[S]_C := \left(S + C\right) \cap \left(S - C\right)</math> be the [[Cone-saturated|<math>C</math>-saturated]] hull of <math>S \subseteq X</math> and for any collection <math>\mathcal{S}</math> of subsets of <math>X</math> let <math>\left[ \mathcal{S} \right]_C := \left\{ \left[ S \right]_C : S \in \mathcal{S} \right\}.</math> 
If <math>C</math> is a cone in a TVS <math>X</math> then <math>C</math> is '''normal''' if <math>\mathcal{U} = \left[ \mathcal{U} \right]_C,</math> where <math>\mathcal{U}</math> is the neighborhood filter at the origin.{{sfn|Schaefer|Wolff|1999|pp=215–222}}

If <math>\mathcal{T}</math> is a collection of subsets of <math>X</math> and if <math>\mathcal{F}</math> is a subset of <math>\mathcal{T}</math> then <math>\mathcal{F}</math> is a '''fundamental subfamily''' of <math>\mathcal{T}</math> if every <math>T \in \mathcal{T}</math> is contained as a subset of some element of <math>\mathcal{F}.</math> 
If <math>\mathcal{G}</math> is a family of subsets of a TVS <math>X</math> then a cone <math>C</math> in <math>X</math> is called a '''<math>\mathcal{G}</math>-cone''' if <math>\left\{ \overline{\left[ G \right]_C} : G \in \mathcal{G} \right\}</math> is a fundamental subfamily of <math>\mathcal{G}</math> and <math>C</math> is a '''strict <math>\mathcal{G}</math>-cone''' if <math>\left\{ \left[ G \right]_C : G \in \mathcal{G} \right\}</math> is a fundamental subfamily of <math>\mathcal{G}.</math>{{sfn|Schaefer|Wolff|1999|pp=215–222}} 
Let <math>\mathcal{B}</math> denote the family of all bounded subsets of <math>X.</math>

If <math>C</math> is a cone in a TVS <math>X</math> (over the real or complex numbers), then the following are equivalent:{{sfn|Schaefer|Wolff|1999|pp=215–222}} 
<ol>
<li> <math>C</math> is a normal cone.</li>
<li> For every filter <math>\mathcal{F}</math> in <math>X,</math> if <math>\lim \mathcal{F} = 0</math> then <math>\lim \left[ \mathcal{F} \right]_C = 0.</math></li>
<li> There exists a neighborhood base <math>\mathcal{G}</math> in <math>X</math> such that <math>B \in \mathcal{G}</math> implies <math>\left[ B \cap C \right]_C \subseteq B.</math></ol>
and if <math>X</math> is a vector space over the reals then we may add to this list:{{sfn|Schaefer|Wolff|1999|pp=215–222}}
<ol start=4>
<li> There exists a neighborhood base at the origin consisting of convex, [[Balanced set|balanced]], [[Cone-saturated|<math>C</math>-saturated]] sets.</li>
<li> There exists a generating family <math>\mathcal{P}</math> of semi-norms on <math>X</math> such that <math>p(x) \leq p(x + y)</math> for all <math>x, y \in C</math> and <math>p \in \mathcal{P}.</math></li>
</ol>
and if <math>X</math> is a locally convex space and if the dual cone of <math>C</math> is denoted by <math>X^{\prime}</math> then we may add to this list:{{sfn|Schaefer|Wolff|1999|pp=215–222}} 
<ol start=6>
<li>For any equicontinuous subset <math>S \subseteq X^{\prime},</math> there exists an equicontiuous <math>B \subseteq C^{\prime}</math> such that <math>S \subseteq B - B.</math></li>
<li>The topology of <math>X</math> is the topology of uniform convergence on the equicontinuous subsets of <math>C^{\prime}.</math></li>
</ol>
and if <math>X</math> is an infrabarreled locally convex space and if <math>\mathcal{B}^{\prime}</math> is the family of all strongly bounded subsets of <math>X^{\prime}</math> then we may add to this list:{{sfn|Schaefer|Wolff|1999|pp=215–222}} 
<ol start=8>
<li>The topology of <math>X</math> is the topology of uniform convergence on strongly bounded subsets of <math>C^{\prime}.</math></li>
<li><math>C^{\prime}</math> is a <math>\mathcal{B}^{\prime}</math>-cone in <math>X^{\prime}.</math>
* this means that the family <math>\left\{ \overline{\left[ B^{\prime} \right]_C} : B^{\prime} \in \mathcal{B}^{\prime} \right\}</math> is a fundamental subfamily of <math>\mathcal{B}^{\prime}.</math>
</li>
<li><math>C^{\prime}</math> is a strict <math>\mathcal{B}^{\prime}</math>-cone in <math>X^{\prime}.</math>
* this means that the family <math>\left\{ \left[ B^{\prime} \right]_C : B^{\prime} \in \mathcal{B}^{\prime} \right\}</math> is a fundamental subfamily of <math>\mathcal{B}^{\prime}.</math>
</li>
</ol>
and if <math>X</math> is an ordered locally convex TVS over the reals whose positive cone is <math>C,</math> then we may add to this list:
<ol start=11>
<li>there exists a Hausdorff locally compact topological space <math>S</math> such that <math>X</math> is isomorphic (as an ordered TVS) with a subspace of <math>R(S),</math> where <math>R(S)</math> is the space of all real-valued continuous functions on <math>X</math> under the topology of compact convergence.{{sfn|Schaefer|Wolff|1999|pp=222-225}} 
</ol>

If <math>X</math> is a locally convex TVS, <math>C</math> is a cone in <math>X</math> with dual cone <math>C^{\prime} \subseteq X^{\prime},</math> and <math>\mathcal{G}</math> is a [[Saturated family|saturated family]] of weakly bounded subsets of <math>X^{\prime},</math> then{{sfn|Schaefer|Wolff|1999|pp=215–222}} 
# if <math>C^{\prime}</math> is a <math>\mathcal{G}</math>-cone then <math>C</math> is a normal cone for the <math>\mathcal{G}</math>-topology on <math>X</math>; 
# if <math>C</math> is a normal cone for a <math>\mathcal{G}</math>-topology on <math>X</math> consistent with <math>\left\langle X, X^{\prime}\right\rangle</math> then <math>C^{\prime}</math> is a strict <math>\mathcal{G}</math>-cone in <math>X^{\prime}.</math>

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

If <math>X</math> is a Banach space and <math>C</math> is a cone in <math>X</math> then the following are equivalent:{{sfn|Schaefer|Wolff|1999|pp=215–222}} 
# <math>C</math> is a <math>\mathcal{B}</math>-cone in <math>X</math>;
# <math>X = \overline{C} - \overline{C}</math>;
# <math>\overline{C}</math> is a strict <math>\mathcal{B}</math>-cone in <math>X.</math>

=== Ordered topological vector spaces ===
Suppose <math>L</math> is an [[Ordered topological vector space|ordered topological vector space]]. That is, <math>L</math> is a [[Topological vector space|topological vector space]], and we define <math>x \geq y</math> whenever <math>x - y</math> lies in the cone <math>L_+</math>. The following statements are equivalent:<ref>{{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}}</ref>

# The cone <math>L_+</math> is normal;
# The normed space <math>L</math> admits an equivalent monotone [[Norm (mathematics)|norm]];
# There exists a constant <math>c > 0</math> such that <math>a \leq x \leq b</math> implies <math>\lVert x \rVert \leq c \max\{\lVert a \rVert, \lVert b \rVert\}</math>;
# The full hull <math>[U] = (U + L_+) \cap (U - L_+)</math> of the closed unit ball <math>U</math> of <math>L</math> is norm bounded;
# There is a constant <math>c > 0</math> such that <math>0 \leq x \leq y</math> implies <math>\lVert x \rVert \leq c \lVert y \rVert</math>.

== Properties ==

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

== Sufficient conditions ==

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

Suppose that <math>\left\{ X_{\alpha} : \alpha \in A \right\}</math> is a family of locally convex TVSs and that <math>C_\alpha</math> is a cone in <math>X_{\alpha}.</math>
If <math>X := \bigoplus_{\alpha} X_{\alpha}</math> is the locally convex direct sum then the cone <math>C := \bigoplus_{\alpha} C_\alpha</math> is a normal cone in <math>X</math> if and only if each <math>X_{\alpha}</math> is normal in <math>X_{\alpha}.</math>{{sfn|Schaefer|Wolff|1999|pp=215–222}}

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

If <math>C</math> is a cone in a locally convex TVS <math>X</math> and if <math>C^{\prime}</math> is the dual cone of <math>C,</math> then <math>X^{\prime} = C^{\prime} - C^{\prime}</math> if and only if <math>C</math> 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 <math>X</math> and <math>Y</math> are ordered locally convex TVSs and if <math>\mathcal{G}</math> is a family of bounded subsets of <math>X,</math> then if the positive cone of <math>X</math> is a <math>\mathcal{G}</math>-cone in <math>X</math> and if the positive cone of <math>Y</math> is a normal cone in <math>Y</math> then the positive cone of <math>L_{\mathcal{G}}(X; Y)</math> is a normal cone for the <math>\mathcal{G}</math>-topology on <math>L(X; Y).</math>{{sfn|Schaefer|Wolff|1999|pp=225-229}}

== See also ==

* {{annotated link|Cone-saturated}}
* {{annotated link|Topological vector lattice}}

== References ==

{{reflist}}

==Bibliography==

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->

{{Functional analysis}}

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