Normal cone (functional analysis)

<h2 id="characterizations">Characterizations</h2>
If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a cone in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then for any subset 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 let 
 
 
 
 [
 S
 
 ]
 
 C
 
 
 :=
 
 (
 
 S
 +
 C
 
 )
 
 ∩
 
 (
 
 S
 −
 C
 
 )
 
 
 
 {\displaystyle [S]_{C}:=\left(S+C\right)\cap \left(S-C\right)}
 
 be the <a href="/facts/Cone-saturated/sIaDXRLu">
 
 
 
 C
 
 
 {\displaystyle C}
 
-saturated</a> hull of 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 and for any collection 
 
 
 
 
 
 S
 
 
 
 
 {\displaystyle {\mathcal {S}}}
 
 of subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 let 
 
 
 
 
 
 [
 
 
 S
 
 
 ]
 
 
 C
 
 
 :=
 
 {
 
 
 
 [
 S
 ]
 
 
 C
 
 
 :
 S
 ∈
 
 
 S
 
 
 
 }
 
 .
 
 
 {\displaystyle \left[{\mathcal {S}}\right]_{C}:=\left\{\left[S\right]_{C}:S\in {\mathcal {S}}\right\}.}
 
 
If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a cone in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 C
 
 
 {\displaystyle 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.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
If 
 
 
 
 
 
 T
 
 
 
 
 {\displaystyle {\mathcal {T}}}
 
 is a collection of subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 is a subset of 
 
 
 
 
 
 T
 
 
 
 
 {\displaystyle {\mathcal {T}}}
 
 then 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 is a fundamental subfamily of 
 
 
 
 
 
 T
 
 
 
 
 {\displaystyle {\mathcal {T}}}
 
 if every 
 
 
 
 T
 ∈
 
 
 T
 
 
 
 
 {\displaystyle T\in {\mathcal {T}}}
 
 is contained as a subset of some element of 
 
 
 
 
 
 F
 
 
 .
 
 
 {\displaystyle {\mathcal {F}}.}
 
 
If 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
 is a family of subsets of a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then a cone 
 
 
 
 C
 
 
 {\displaystyle C}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is called a 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-cone if 
 
 
 
 
 {
 
 
 
 
 
 [
 G
 ]
 
 
 C
 
 
 ¯
 
 
 :
 G
 ∈
 
 
 G
 
 
 
 }
 
 
 
 {\displaystyle \left\{{\overline {\left[G\right]_{C}}}:G\in {\mathcal {G}}\right\}}
 
 is a fundamental subfamily of 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
 and 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a strict 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-cone if 
 
 
 
 
 {
 
 
 
 [
 G
 ]
 
 
 C
 
 
 :
 G
 ∈
 
 
 G
 
 
 
 }
 
 
 
 {\displaystyle \left\{\left[G\right]_{C}:G\in {\mathcal {G}}\right\}}
 
 is a fundamental subfamily of 
 
 
 
 
 
 G
 
 
 .
 
 
 {\displaystyle {\mathcal {G}}.}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> 
Let 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 denote the family of all bounded subsets of 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a cone in a TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (over the real or complex numbers), then the following are equivalent:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> 

<ol>
<li> 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a normal cone.</li>
<li> For every filter 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 if 
 
 
 
 lim
 
 
 F
 
 
 =
 0
 
 
 {\displaystyle \lim {\mathcal {F}}=0}
 
 then 
 
 
 
 lim
 
 
 [
 
 
 F
 
 
 ]
 
 
 C
 
 
 =
 0.
 
 
 {\displaystyle \lim \left[{\mathcal {F}}\right]_{C}=0.}
 
</li>
<li> There exists a neighborhood base 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 B
 ∈
 
 
 G
 
 
 
 
 {\displaystyle B\in {\mathcal {G}}}
 
 implies 
 
 
 
 
 
 [
 
 B
 ∩
 C
 
 ]
 
 
 C
 
 
 ⊆
 B
 .
 
 
 {\displaystyle \left[B\cap C\right]_{C}\subseteq B.}
 
</li></ol>
and if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a vector space over the reals then we may add to this list:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<ol>
<li> There exists a neighborhood base at the origin consisting of convex, <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a>, <a href="/facts/Cone-saturated/sIaDXRLu">
 
 
 
 C
 
 
 {\displaystyle C}
 
-saturated</a> sets.</li>
<li> There exists a generating family 
 
 
 
 
 
 P
 
 
 
 
 {\displaystyle {\mathcal {P}}}
 
 of semi-norms on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that 
 
 
 
 p
 (
 x
 )
 ≤
 p
 (
 x
 +
 y
 )
 
 
 {\displaystyle p(x)\leq p(x+y)}
 
 for all 
 
 
 
 x
 ,
 y
 ∈
 C
 
 
 {\displaystyle x,y\in C}
 
 and 
 
 
 
 p
 ∈
 
 
 P
 
 
 .
 
 
 {\displaystyle p\in {\mathcal {P}}.}
 
</li>
</ol>
and if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a locally convex space and if the dual cone of 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is denoted by 
 
 
 
 
 X
 
 ′
 
 
 
 
 {\displaystyle X^{\prime }}
 
 then we may add to this list:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 

<ol>
<li>For any equicontinuous subset 
 
 
 
 S
 ⊆
 
 X
 
 ′
 
 
 ,
 
 
 {\displaystyle S\subseteq X^{\prime },}
 
 there exists an equicontiuous 
 
 
 
 B
 ⊆
 
 C
 
 ′
 
 
 
 
 {\displaystyle B\subseteq C^{\prime }}
 
 such that 
 
 
 
 S
 ⊆
 B
 −
 B
 .
 
 
 {\displaystyle S\subseteq B-B.}
 
</li>
<li>The topology of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the topology of uniform convergence on the equicontinuous subsets of 
 
 
 
 
 C
 
 ′
 
 
 .
 
 
 {\displaystyle C^{\prime }.}
 
</li>
</ol>
and if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is an <a href="/facts/Infrabarreled/SaoLAH83">infrabarreled</a> locally convex space and if 
 
 
 
 
 
 
 B
 
 
 
 ′
 
 
 
 
 {\displaystyle {\mathcal {B}}^{\prime }}
 
 is the family of all strongly bounded subsets of 
 
 
 
 
 X
 
 ′
 
 
 
 
 {\displaystyle X^{\prime }}
 
 then we may add to this list:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> 

<ol>
<li>The topology of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the topology of uniform convergence on strongly bounded subsets of 
 
 
 
 
 C
 
 ′
 
 
 .
 
 
 {\displaystyle C^{\prime }.}
 
</li>
<li>
 
 
 
 
 C
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }}
 
 is a 
 
 
 
 
 
 
 B
 
 
 
 ′
 
 
 
 
 {\displaystyle {\mathcal {B}}^{\prime }}
 
-cone in 
 
 
 
 
 X
 
 ′
 
 
 .
 
 
 {\displaystyle X^{\prime }.}

<ul><li>this means that the family 
 
 
 
 
 {
 
 
 
 
 
 [
 
 B
 
 ′
 
 
 ]
 
 
 C
 
 
 ¯
 
 
 :
 
 B
 
 ′
 
 
 ∈
 
 
 
 B
 
 
 
 ′
 
 
 
 }
 
 
 
 {\displaystyle \left\{{\overline {\left[B^{\prime }\right]_{C}}}:B^{\prime }\in {\mathcal {B}}^{\prime }\right\}}
 
 is a fundamental subfamily of 
 
 
 
 
 
 
 B
 
 
 
 ′
 
 
 .
 
 
 {\displaystyle {\mathcal {B}}^{\prime }.}
 
</li></ul>
</li>
<li>
 
 
 
 
 C
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }}
 
 is a strict 
 
 
 
 
 
 
 B
 
 
 
 ′
 
 
 
 
 {\displaystyle {\mathcal {B}}^{\prime }}
 
-cone in 
 
 
 
 
 X
 
 ′
 
 
 .
 
 
 {\displaystyle X^{\prime }.}

<ul><li>this means that the family 
 
 
 
 
 {
 
 
 
 [
 
 B
 
 ′
 
 
 ]
 
 
 C
 
 
 :
 
 B
 
 ′
 
 
 ∈
 
 
 
 B
 
 
 
 ′
 
 
 
 }
 
 
 
 {\displaystyle \left\{\left[B^{\prime }\right]_{C}:B^{\prime }\in {\mathcal {B}}^{\prime }\right\}}
 
 is a fundamental subfamily of 
 
 
 
 
 
 
 B
 
 
 
 ′
 
 
 .
 
 
 {\displaystyle {\mathcal {B}}^{\prime }.}
 
</li></ul>
</li>
</ol>
and if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is an ordered locally convex TVS over the reals whose positive cone is 
 
 
 
 C
 ,
 
 
 {\displaystyle C,}
 
 then we may add to this list:

<ol>
<li>there exists a Hausdorff <a href="/facts/Locally_compact/hesalT7q">locally compact</a> topological space 
 
 
 
 S
 
 
 {\displaystyle S}
 
 such that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is isomorphic (as an ordered TVS) with a subspace of 
 
 
 
 R
 (
 S
 )
 ,
 
 
 {\displaystyle R(S),}
 
 where 
 
 
 
 R
 (
 S
 )
 
 
 {\displaystyle R(S)}
 
 is the space of all real-valued continuous functions on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 under the topology of compact convergence.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> 
</li></ol>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> TVS, 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a cone in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with <a href="/facts/Dual_cone/2PnBdr7p">dual cone</a> 
 
 
 
 
 C
 
 ′
 
 
 ⊆
 
 X
 
 ′
 
 
 ,
 
 
 {\displaystyle C^{\prime }\subseteq X^{\prime },}
 
 and 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
 is a <a href="/facts/Saturated_family/Rxo5YK28">saturated family</a> of weakly bounded subsets of 
 
 
 
 
 X
 
 ′
 
 
 ,
 
 
 {\displaystyle X^{\prime },}
 
 then<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> 

<ol><li>if 
 
 
 
 
 C
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }}
 
 is a 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-cone then 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a normal cone for the 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
;</li>
<li>if 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a normal cone for a 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 consistent with 
 
 
 
 
 ⟨
 
 X
 ,
 
 X
 
 ′
 
 
 
 ⟩
 
 
 
 {\displaystyle \left\langle X,X^{\prime }\right\rangle }
 
 then 
 
 
 
 
 C
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }}
 
 is a strict 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-cone in 
 
 
 
 
 X
 
 ′
 
 
 .
 
 
 {\displaystyle X^{\prime }.}
 
</li></ol>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a Banach space, 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a closed cone in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
, and 
 
 
 
 
 
 
 B
 
 
 
 ′
 
 
 
 
 {\displaystyle {\mathcal {B}}^{\prime }}
 
 is the family of all bounded subsets of 
 
 
 
 
 X
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle X_{b}^{\prime }}
 
 then the <a href="/facts/Dual_cone/2PnBdr7p">dual cone</a> 
 
 
 
 
 C
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }}
 
 is normal in 
 
 
 
 
 X
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle X_{b}^{\prime }}
 
 if and only if 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a strict 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
-cone.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a Banach space and 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a cone in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then the following are equivalent:<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> 

<ol><li>
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
-cone in 
 
 
 
 X
 
 
 {\displaystyle X}
 
;</li>
<li>
 
 
 
 X
 =
 
 
 C
 ¯
 
 
 −
 
 
 C
 ¯
 
 
 
 
 {\displaystyle X={\overline {C}}-{\overline {C}}}
 
;</li>
<li>
 
 
 
 
 
 C
 ¯
 
 
 
 
 {\displaystyle {\overline {C}}}
 
 is a strict 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
-cone in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
</li></ol>
<h3>Ordered topological vector spaces</h3>
Suppose 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is an <a href="/facts/Ordered_topological_vector_space/N1fvN8th">ordered topological vector space</a>. That is, 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a>, and we define 
 
 
 
 x
 ≥
 y
 
 
 {\displaystyle x\geq y}
 
 whenever 
 
 
 
 x
 −
 y
 
 
 {\displaystyle x-y}
 
 lies in the cone 
 
 
 
 
 L
 
 +
 
 
 
 
 {\displaystyle L_{+}}
 
. The following statements are equivalent:<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<ol><li>The cone 
 
 
 
 
 L
 
 +
 
 
 
 
 {\displaystyle L_{+}}
 
 is normal;</li>
<li>The normed space 
 
 
 
 L
 
 
 {\displaystyle L}
 
 admits an equivalent monotone <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>;</li>
<li>There exists a constant 
 
 
 
 c
 >
 0
 
 
 {\displaystyle c>0}
 
 such that 
 
 
 
 a
 ≤
 x
 ≤
 b
 
 
 {\displaystyle a\leq x\leq b}
 
 implies 
 
 
 
 ‖
 x
 ‖
 ≤
 c
 max
 {
 ‖
 a
 ‖
 ,
 ‖
 b
 ‖
 }
 
 
 {\displaystyle \lVert x\rVert \leq c\max\{\lVert a\rVert ,\lVert b\rVert \}}
 
;</li>
<li>The full hull 
 
 
 
 [
 U
 ]
 =
 (
 U
 +
 
 L
 
 +
 
 
 )
 ∩
 (
 U
 −
 
 L
 
 +
 
 
 )
 
 
 {\displaystyle [U]=(U+L_{+})\cap (U-L_{+})}
 
 of the closed unit ball 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is norm bounded;</li>
<li>There is a constant 
 
 
 
 c
 >
 0
 
 
 {\displaystyle c>0}
 
 such that 
 
 
 
 0
 ≤
 x
 ≤
 y
 
 
 {\displaystyle 0\leq x\leq y}
 
 implies 
 
 
 
 ‖
 x
 ‖
 ≤
 c
 ‖
 y
 ‖
 
 
 {\displaystyle \lVert x\rVert \leq c\lVert y\rVert }
 
.</li></ol>
<h2 id="properties">Properties</h2>
<ul><li>If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a Hausdorff TVS then every normal cone in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a proper cone.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a normable space and if 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a normal cone in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then 
 
 
 
 
 X
 
 ′
 
 
 =
 
 C
 
 ′
 
 
 −
 
 C
 
 ′
 
 
 .
 
 
 {\displaystyle X^{\prime }=C^{\prime }-C^{\prime }.}
 
<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li>
<li>Suppose that the positive cone of an ordered locally convex TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is weakly normal in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and that 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is an ordered locally convex TVS with positive cone 
 
 
 
 D
 .
 
 
 {\displaystyle D.}
 
 If 
 
 
 
 Y
 =
 D
 −
 D
 
 
 {\displaystyle Y=D-D}
 
 then 
 
 
 
 H
 −
 H
 
 
 {\displaystyle H-H}
 
 is dense in 
 
 
 
 
 L
 
 s
 
 
 (
 X
 ;
 Y
 )
 
 
 {\displaystyle L_{s}(X;Y)}
 
 where 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is the canonical positive cone of 
 
 
 
 L
 (
 X
 ;
 Y
 )
 
 
 {\displaystyle L(X;Y)}
 
 and 
 
 
 
 
 L
 
 s
 
 
 (
 X
 ;
 Y
 )
 
 
 {\displaystyle L_{s}(X;Y)}
 
 is the space 
 
 
 
 L
 (
 X
 ;
 Y
 )
 
 
 {\displaystyle L(X;Y)}
 
 with the topology of simple convergence.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
<ul><li>If 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
 is a family of bounded subsets of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then there are apparently no simple conditions guaranteeing that 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is a 
 
 
 
 
 
 T
 
 
 
 
 {\displaystyle {\mathcal {T}}}
 
-cone in 
 
 
 
 
 L
 
 
 G
 
 
 
 (
 X
 ;
 Y
 )
 ,
 
 
 {\displaystyle L_{\mathcal {G}}(X;Y),}
 
 even for the most common types of families 
 
 
 
 
 
 T
 
 
 
 
 {\displaystyle {\mathcal {T}}}
 
 of bounded subsets of 
 
 
 
 
 L
 
 
 G
 
 
 
 (
 X
 ;
 Y
 )
 
 
 {\displaystyle L_{\mathcal {G}}(X;Y)}
 
 (except for very special cases).<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li></ul></li></ul>
<h2 id="sufficient-conditions">Sufficient conditions</h2>
If the topology on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is locally convex then the closure of a normal cone is a normal cone.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
Suppose that 
 
 
 
 
 {
 
 
 X
 
 α
 
 
 :
 α
 ∈
 A
 
 }
 
 
 
 {\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}
 
 is a family of locally convex TVSs and that 
 
 
 
 
 C
 
 α
 
 
 
 
 {\displaystyle C_{\alpha }}
 
 is a cone in 
 
 
 
 
 X
 
 α
 
 
 .
 
 
 {\displaystyle X_{\alpha }.}

If 
 
 
 
 X
 :=
 
 ⨁
 
 α
 
 
 
 X
 
 α
 
 
 
 
 {\displaystyle X:=\bigoplus _{\alpha }X_{\alpha }}
 
 is the locally convex direct sum then the cone 
 
 
 
 C
 :=
 
 ⨁
 
 α
 
 
 
 C
 
 α
 
 
 
 
 {\displaystyle C:=\bigoplus _{\alpha }C_{\alpha }}
 
 is a normal cone in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if and only if each 
 
 
 
 
 X
 
 α
 
 
 
 
 {\displaystyle X_{\alpha }}
 
 is normal in 
 
 
 
 
 X
 
 α
 
 
 .
 
 
 {\displaystyle X_{\alpha }.}
 
<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a locally convex space then the closure of a normal cone is a normal cone.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a cone in a locally convex TVS 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if 
 
 
 
 
 C
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }}
 
 is the dual cone of 
 
 
 
 C
 ,
 
 
 {\displaystyle C,}
 
 then 
 
 
 
 
 X
 
 ′
 
 
 =
 
 C
 
 ′
 
 
 −
 
 C
 
 ′
 
 
 
 
 {\displaystyle X^{\prime }=C^{\prime }-C^{\prime }}
 
 if and only if 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is weakly normal.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> 
Every normal cone in a locally convex TVS is weakly normal.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> 
In a normed space, a cone is normal if and only if it is weakly normal.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are ordered locally convex TVSs and if 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
 is a family of bounded subsets of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 then if the positive cone of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-cone in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and if the positive cone of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a normal cone in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 then the positive cone of 
 
 
 
 
 L
 
 
 G
 
 
 
 (
 X
 ;
 Y
 )
 
 
 {\displaystyle L_{\mathcal {G}}(X;Y)}
 
 is a normal cone for the 
 
 
 
 
 
 G
 
 
 
 
 {\displaystyle {\mathcal {G}}}
 
-topology on 
 
 
 
 L
 (
 X
 ;
 Y
 )
 .
 
 
 {\displaystyle L(X;Y).}
 
<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cone-saturated/sIaDXRLu">Cone-saturated</a></li>
<li><a href="/facts/Topological_vector_lattice/q5BLo3cr">Topological vector lattice</a></li>
<li><a href="/facts/Vector_lattice/FX7Mwfxe">Vector lattice</a> – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets</li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Schaefer & Wolff 1999, pp. 222–225. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Aliprantis, Charalambos D. (2007). Cones and duality. Rabee Tourky. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-4146-4. OCLC 87808043. <a href="978-0-8218-4146-4" target="_blank">978-0-8218-4146-4</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Schaefer & Wolff 1999, pp. 225–229. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Schaefer & Wolff 1999, pp. 225–229. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Schaefer & Wolff 1999, pp. 215–222. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Schaefer & Wolff 1999, pp. 225–229. - 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. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
</ol>

Normal cone (functional analysis) open-in-new

Normal cone (functional analysis)