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 ${\displaystyle {\mathcal {U))}$ is the neighborhood filter at the origin, then ${\displaystyle C}$ is called normal if ${\displaystyle {\mathcal {U))=\left[{\mathcal {U))\right]_{C},}$ where ${\displaystyle \left[{\mathcal {U))\right]_{C}:=\left\{[U]_{C}:U\in {\mathcal {U))\right\))$ 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.

Characterizations

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

If ${\displaystyle {\mathcal {T))}$ is a collection of subsets of ${\displaystyle X}$ and if ${\displaystyle {\mathcal {F))}$ is a subset of ${\displaystyle {\mathcal {T))}$ then ${\displaystyle {\mathcal {F))}$ is a fundamental subfamily of ${\displaystyle {\mathcal {T))}$ if every ${\displaystyle T\in {\mathcal {T))}$ is contained as a subset of some element of ${\displaystyle {\mathcal {F)).}$ If ${\displaystyle {\mathcal {G))}$ is a family of subsets of a TVS ${\displaystyle X}$ then a cone ${\displaystyle C}$ in ${\displaystyle X}$ is called a ${\displaystyle {\mathcal {G))}$-cone if ${\displaystyle \left\((\overline {\left[G\right]_{C))}:G\in {\mathcal {G))\right\))$ is a fundamental subfamily of ${\displaystyle {\mathcal {G))}$ and ${\displaystyle C}$ is a strict ${\displaystyle {\mathcal {G))}$-cone if ${\displaystyle \left\{\left[G\right]_{C}:G\in {\mathcal {G))\right\))$ is a fundamental subfamily of ${\displaystyle {\mathcal {G)).}$^[1] Let ${\displaystyle {\mathcal {B))}$ 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 ${\displaystyle {\mathcal {F))}$ in ${\displaystyle X,}$ if ${\displaystyle \lim {\mathcal {F))=0}$ then ${\displaystyle \lim \left[{\mathcal {F))\right]_{C}=0.}$
3. There exists a neighborhood base ${\displaystyle {\mathcal {G))}$ in ${\displaystyle X}$ such that ${\displaystyle B\in {\mathcal {G))}$ implies ${\displaystyle \left[B\cap C\right]_{C}\subseteq B.}$

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 ${\displaystyle {\mathcal {P))}$ of semi-norms on ${\displaystyle X}$ such that ${\displaystyle p(x)\leq p(x+y)}$ for all ${\displaystyle x,y\in C}$ and ${\displaystyle p\in {\mathcal {P)).}$

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

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 ${\displaystyle {\mathcal {G))}$ is a saturated family of weakly bounded subsets of ${\displaystyle X^{\prime },}$ then^[1]

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

If ${\displaystyle X}$ is a Banach space, ${\displaystyle C}$ is a closed cone in ${\displaystyle X,}$, and ${\displaystyle {\mathcal {B))^{\prime ))$ 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 ${\displaystyle {\mathcal {B))}$-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 ${\displaystyle {\mathcal {B))}$-cone in ${\displaystyle X}$;
2. ${\displaystyle X={\overline {C))-{\overline {C))}$;
3. ${\displaystyle {\overline {C))}$ is a strict ${\displaystyle {\mathcal {B))}$-cone in ${\displaystyle X.}$

Ordered topological vector spaces

Suppose ${\displaystyle L}$ is an ordered topological vector space. That is, ${\displaystyle L}$ is a topological vector space, and we define ${\displaystyle x\geq y}$ 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 ${\displaystyle a\leq x\leq b}$ implies ${\displaystyle \lVert x\rVert \leq c\max\{\lVert a\rVert ,\lVert b\rVert \))$;
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 ${\displaystyle 0\leq x\leq y}$ implies ${\displaystyle \lVert x\rVert \leq c\lVert y\rVert }$.

Properties

• 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 ${\displaystyle {\mathcal {G))}$ is a family of bounded subsets of ${\displaystyle X,}$ then there are apparently no simple conditions guaranteeing that ${\displaystyle H}$ is a ${\displaystyle {\mathcal {T))}$-cone in ${\displaystyle L_{\mathcal {G))(X;Y),}$ even for the most common types of families ${\displaystyle {\mathcal {T))}$ of bounded subsets of ${\displaystyle L_{\mathcal {G))(X;Y)}$ (except for very special cases).^[4]

Sufficient conditions

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

Suppose that ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\))$ is a family of locally convex TVSs and that ${\displaystyle C_{\alpha ))$ is a cone in ${\displaystyle X_{\alpha }.}$ If ${\displaystyle X:=\bigoplus _{\alpha }X_{\alpha ))$ is the locally convex direct sum then the cone ${\displaystyle C:=\bigoplus _{\alpha }C_{\alpha ))$ 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 ${\displaystyle {\mathcal {G))}$ is a family of bounded subsets of ${\displaystyle X,}$ then if the positive cone of ${\displaystyle X}$ is a ${\displaystyle {\mathcal {G))}$-cone in ${\displaystyle X}$ and if the positive cone of ${\displaystyle Y}$ is a normal cone in ${\displaystyle Y}$ then the positive cone of ${\displaystyle L_{\mathcal {G))(X;Y)}$ is a normal cone for the ${\displaystyle {\mathcal {G))}$-topology on ${\displaystyle L(X;Y).}$^[4]