Normal cone (functional analysis)

In mathematics, specifically in order theory and functional analysis, if ${\displaystyle C}$ is a cone at the origin in a topological vector space ${\displaystyle X}$ such that ${\displaystyle 0\in C}$ and if ${\displaystyle {𝒰}}$ is the neighborhood filter at the origin, then ${\displaystyle C}$ is called normal if ${\displaystyle {𝒰}={\left[{𝒰}\right]}_C,}$ where ${\displaystyle {\left[{𝒰}\right]}_C:=\{[U{]}_C:U\in {𝒰}\}}$ and where for any subset ${\displaystyle S\subseteq X,}$ ${\displaystyle [S{]}_C:=(S+C)\cap (S-C)}$ is the ${\displaystyle C}$-saturatation of ${\displaystyle S.}$^[1]

Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.

If ${\displaystyle C}$ is a cone in a TVS ${\displaystyle X}$ then for any subset ${\displaystyle S\subseteq X}$ let ${\displaystyle [S{]}_C:=(S+C)\cap (S-C)}$ be the ${\displaystyle C}$-saturated hull of ${\displaystyle S\subseteq X}$ and for any collection ${\displaystyle {𝒮}}$ of subsets of ${\displaystyle X}$ let ${\displaystyle {\left[{𝒮}\right]}_C:=\{{\left[S\right]}_C:S\in {𝒮}\}.}$ If ${\displaystyle C}$ is a cone in a TVS ${\displaystyle X}$ then ${\displaystyle C}$ is normal if ${\displaystyle {𝒰}={\left[{𝒰}\right]}_C,}$ where ${\displaystyle {𝒰}}$ is the neighborhood filter at the origin.^[1]

If ${\displaystyle {𝒯}}$ is a collection of subsets of ${\displaystyle X}$ and if ${\displaystyle {ℱ}}$ is a subset of ${\displaystyle {𝒯}}$ then ${\displaystyle {ℱ}}$ is a fundamental subfamily of ${\displaystyle {𝒯}}$ if every ${\displaystyle T\in {𝒯}}$ is contained as a subset of some element of ${\displaystyle {ℱ}.}$ If ${\displaystyle {𝒢}}$ is a family of subsets of a TVS ${\displaystyle X}$ then a cone ${\displaystyle C}$ in ${\displaystyle X}$ is called a ${\displaystyle {𝒢}}$-cone if ${\displaystyle \{\overline{{\left[G\right]}_C}:G\in {𝒢}\}}$ is a fundamental subfamily of ${\displaystyle {𝒢}}$ and ${\displaystyle C}$ is a strict ${\displaystyle {𝒢}}$-cone if ${\displaystyle \{{\left[G\right]}_C:G\in {𝒢}\}}$ is a fundamental subfamily of ${\displaystyle {𝒢}.}$^[1] Let ${\displaystyle {ℬ}}$ denote the family of all bounded subsets of ${\displaystyle X.}$

If ${\displaystyle C}$ is a cone in a TVS ${\displaystyle X}$ (over the real or complex numbers), then the following are equivalent:^[1]

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

and if ${\displaystyle X}$ is a vector space over the reals then we may add to this list:^[1]

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

and if ${\displaystyle X}$ is a locally convex space and if the dual cone of ${\displaystyle C}$ is denoted by ${\displaystyle X^{\prime }}$ then we may add to this list:^[1]

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

and if ${\displaystyle X}$ is an infrabarreled locally convex space and if ${\displaystyle {{ℬ}}^{\prime }}$ is the family of all strongly bounded subsets of ${\displaystyle X^{\prime }}$ then we may add to this list:^[1]

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

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

11. there exists a Hausdorff locally compact topological space ${\displaystyle S}$ such that ${\displaystyle X}$ is isomorphic (as an ordered TVS) with a subspace of ${\displaystyle R(S),}$ where ${\displaystyle R(S)}$ is the space of all real-valued continuous functions on ${\displaystyle X}$ under the topology of compact convergence.^[2]

If ${\displaystyle X}$ is a locally convex TVS, ${\displaystyle C}$ is a cone in ${\displaystyle X}$ with dual cone ${\displaystyle C^{\prime }\subseteq X^{\prime },}$ and ${\displaystyle {𝒢}}$ is a saturated family of weakly bounded subsets of ${\displaystyle X^{\prime },}$ then^[1]

1. if ${\displaystyle C^{\prime }}$ is a ${\displaystyle {𝒢}}$-cone then ${\displaystyle C}$ is a normal cone for the ${\displaystyle {𝒢}}$-topology on ${\displaystyle X}$;
2. if ${\displaystyle C}$ is a normal cone for a ${\displaystyle {𝒢}}$-topology on ${\displaystyle X}$ consistent with ${\displaystyle ⟨X,X^{\prime }⟩}$ then ${\displaystyle C^{\prime }}$ is a strict ${\displaystyle {𝒢}}$-cone in ${\displaystyle X^{\prime }.}$

If ${\displaystyle X}$ is a Banach space, ${\displaystyle C}$ is a closed cone in ${\displaystyle X,}$, and ${\displaystyle {{ℬ}}^{\prime }}$ is the family of all bounded subsets of ${\displaystyle X_b^{\prime }}$ then the dual cone ${\displaystyle C^{\prime }}$ is normal in ${\displaystyle X_b^{\prime }}$ if and only if ${\displaystyle C}$ is a strict ${\displaystyle {ℬ}}$-cone.^[1]

If ${\displaystyle X}$ is a Banach space and ${\displaystyle C}$ is a cone in ${\displaystyle X}$ then the following are equivalent:^[1]

1. ${\displaystyle C}$ is a ${\displaystyle {ℬ}}$-cone in ${\displaystyle X}$;
2. ${\displaystyle X=\bar{C}-\bar{C}}$;
3. ${\displaystyle \bar{C}}$ is a strict ${\displaystyle {ℬ}}$-cone in ${\displaystyle X.}$

Suppose ${\displaystyle L}$ is an ordered topological vector space. That is, ${\displaystyle L}$ is a topological vector space, and we define ${\displaystyle x\ge y}$ whenever ${\displaystyle x-y}$ lies in the cone ${\displaystyle L_{+}}$. The following statements are equivalent:^[3]

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

• If ${\displaystyle X}$ is a Hausdorff TVS then every normal cone in ${\displaystyle X}$ is a proper cone.^[1]
• If ${\displaystyle X}$ is a normable space and if ${\displaystyle C}$ is a normal cone in ${\displaystyle X}$ then ${\displaystyle X^{\prime }=C^{\prime }-C^{\prime }.}$^[1]
• Suppose that the positive cone of an ordered locally convex TVS ${\displaystyle X}$ is weakly normal in ${\displaystyle X}$ and that ${\displaystyle Y}$ is an ordered locally convex TVS with positive cone ${\displaystyle D.}$ If ${\displaystyle Y=D-D}$ then ${\displaystyle H-H}$ is dense in ${\displaystyle L_s(X;Y)}$ where ${\displaystyle H}$ is the canonical positive cone of ${\displaystyle L(X;Y)}$ and ${\displaystyle L_s(X;Y)}$ is the space ${\displaystyle L(X;Y)}$ with the topology of simple convergence.^[4]
  • If ${\displaystyle {𝒢}}$ is a family of bounded subsets of ${\displaystyle X,}$ then there are apparently no simple conditions guaranteeing that ${\displaystyle H}$ is a ${\displaystyle {𝒯}}$-cone in ${\displaystyle L_{{𝒢}}(X;Y),}$ even for the most common types of families ${\displaystyle {𝒯}}$ of bounded subsets of ${\displaystyle L_{{𝒢}}(X;Y)}$ (except for very special cases).^[4]

If the topology on ${\displaystyle X}$ is locally convex then the closure of a normal cone is a normal cone.^[1]

Suppose that ${\displaystyle \{X_{\alpha }:\alpha \in A\}}$ is a family of locally convex TVSs and that ${\displaystyle C_{\alpha }}$ is a cone in ${\displaystyle X_{\alpha }.}$ If ${\displaystyle X:=\underset{\alpha }{⨁}{X}_{\alpha }}$ is the locally convex direct sum then the cone ${\displaystyle C:=\underset{\alpha }{⨁}{C}_{\alpha }}$ is a normal cone in ${\displaystyle X}$ if and only if each ${\displaystyle X_{\alpha }}$ is normal in ${\displaystyle X_{\alpha }.}$^[1]

If ${\displaystyle X}$ is a locally convex space then the closure of a normal cone is a normal cone.^[1]

If ${\displaystyle C}$ is a cone in a locally convex TVS ${\displaystyle X}$ and if ${\displaystyle C^{\prime }}$ is the dual cone of ${\displaystyle C,}$ then ${\displaystyle X^{\prime }=C^{\prime }-C^{\prime }}$ if and only if ${\displaystyle C}$ is weakly normal.^[1] Every normal cone in a locally convex TVS is weakly normal.^[1] In a normed space, a cone is normal if and only if it is weakly normal.^[1]

If ${\displaystyle X}$ and ${\displaystyle Y}$ are ordered locally convex TVSs and if ${\displaystyle {𝒢}}$ is a family of bounded subsets of ${\displaystyle X,}$ then if the positive cone of ${\displaystyle X}$ is a ${\displaystyle {𝒢}}$-cone in ${\displaystyle X}$ and if the positive cone of ${\displaystyle Y}$ is a normal cone in ${\displaystyle Y}$ then the positive cone of ${\displaystyle L_{{𝒢}}(X;Y)}$ is a normal cone for the ${\displaystyle {𝒢}}$-topology on ${\displaystyle L(X;Y).}$^[4]

• Cone-saturated
• Topological vector lattice

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 

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