Normal cone (functional analysis)

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

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

Characterizations

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

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

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

and if $X$ is a vector space over the reals then we may add to this list:

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

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

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

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

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

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.

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

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

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

If $X$ is a Banach space and $C$ is a cone in $X$ then the following are equivalent:

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

Ordered topological vector spaces

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

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 $a\le x\le b$ implies $\Vert x\Vert \le c\max \{\Vert a\Vert ,\Vert b\Vert \}$;
4. The full hull $[U]=(U+L_{+})\cap (U-L_{+})$ of the closed unit ball $U$ of $L$ is norm bounded;
5. There is a constant $c>0$ such that $0\le x\le y$ implies $\Vert x\Vert \le c\Vert y\Vert$.

Properties

• If $X$ is a Hausdorff TVS then every normal cone in $X$ is a proper cone.
• If $X$ is a normable space and if $C$ is a normal cone in $X$ then $X^{\prime }=C^{\prime }-C^{\prime }.$
• 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.
  • If ${𝒢}$ is a family of bounded subsets of $X,$ then there are apparently no simple conditions guaranteeing that $H$ is a ${𝒯}$-cone in $L_{{𝒢}}(X;Y),$ even for the most common types of families ${𝒯}$ of bounded subsets of $L_{{𝒢}}(X;Y)$ (except for very special cases).

Sufficient conditions

If the topology on $X$ is locally convex then the closure of a normal cone is a normal cone.

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

If $X$ is a locally convex space then the closure of a normal cone is a normal cone.

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

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

See also

• Cone-saturated
• Topological vector lattice

Bibliography

• 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.