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

If $$\mathcal{T}$$ is a collection of subsets of $$X$$ and if $$\mathcal{F}$$ is a subset of $$\mathcal{T}$$ then $$\mathcal{F}$$ is a fundamental subfamily of $$\mathcal{T}$$ if every $$T \in \mathcal{T}$$ is contained as a subset of some element of $$\mathcal{F}.$$ If $$\mathcal{G}$$ is a family of subsets of a TVS $$X$$ then a cone $$C$$ in $$X$$ is called a $$\mathcal{G}$$-cone if $$\left\{ \overline{\left[ G \right]_C} : G \in \mathcal{G} \right\}$$ is a fundamental subfamily of $$\mathcal{G}$$ and $$C$$ is a strict $$\mathcal{G}$$-cone if $$\left\{ \left[ G \right]_C : G \in \mathcal{G} \right\}$$ is a fundamental subfamily of $$\mathcal{G}.$$ Let $$\mathcal{B}$$ 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:   $$C$$ is a normal cone.  For every filter $$\mathcal{F}$$ in $$X,$$ if $$\lim \mathcal{F} = 0$$ then $$\lim \left[ \mathcal{F} \right]_C = 0.$$  There exists a neighborhood base $$\mathcal{G}$$ in $$X$$ such that $$B \in \mathcal{G}$$ implies $$\left[ B \cap C \right]_C \subseteq B.$$ and if $$X$$ is a vector space over the reals then we may add to this list:  There exists a neighborhood base at the origin consisting of convex, balanced, $C$-saturated sets.  There exists a generating family $$\mathcal{P}$$ of semi-norms on $$X$$ such that $$p(x) \leq p(x + y)$$ for all $$x, y \in C$$ and $$p \in \mathcal{P}.$$  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: For any equicontinuous subset $$S \subseteq X^{\prime},$$ there exists an equicontiuous $$B \subseteq C^{\prime}$$ such that $$S \subseteq B - B.$$ The topology of $$X$$ is the topology of uniform convergence on the equicontinuous subsets of $$C^{\prime}.$$ </ol> and if $$X$$ is an infrabarreled locally convex space and if $$\mathcal{B}^{\prime}$$ is the family of all strongly bounded subsets of $$X^{\prime}$$ then we may add to this list: <li>The topology of $$X$$ is the topology of uniform convergence on strongly bounded subsets of $$C^{\prime}.$$</li> <li>$$C^{\prime}$$ is a $$\mathcal{B}^{\prime}$$-cone in $$X^{\prime}.$$ </li> <li>$$C^{\prime}$$ is a strict $$\mathcal{B}^{\prime}$$-cone in $$X^{\prime}.$$ </li> </ol> and if $$X$$ is an ordered locally convex TVS over the reals whose positive cone is $$C,$$ then we may add to this list: <li>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. </ol>
 * this means that the family $$\left\{ \overline{\left[ B^{\prime} \right]_C} : B^{\prime} \in \mathcal{B}^{\prime} \right\}$$ is a fundamental subfamily of $$\mathcal{B}^{\prime}.$$
 * this means that the family $$\left\{ \left[ B^{\prime} \right]_C : B^{\prime} \in \mathcal{B}^{\prime} \right\}$$ is a fundamental subfamily of $$\mathcal{B}^{\prime}.$$

If $$X$$ is a locally convex TVS, $$C$$ is a cone in $$X$$ with dual cone $$C^{\prime} \subseteq X^{\prime},$$ and $$\mathcal{G}$$ is a saturated family of weakly bounded subsets of $$X^{\prime},$$ then
 * 1) if $$C^{\prime}$$ is a $$\mathcal{G}$$-cone then $$C$$ is a normal cone for the $$\mathcal{G}$$-topology on $$X$$;
 * 2) if $$C$$ is a normal cone for a $$\mathcal{G}$$-topology on $$X$$ consistent with $$\left\langle X, X^{\prime}\right\rangle$$ then $$C^{\prime}$$ is a strict $$\mathcal{G}$$-cone in $$X^{\prime}.$$

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

If $$X$$ is a Banach space and $$C$$ is a cone in $$X$$ then the following are equivalent:
 * 1) $$C$$ is a $$\mathcal{B}$$-cone in $$X$$;
 * 2) $$X = \overline{C} - \overline{C}$$;
 * 3) $$\overline{C}$$ is a strict $$\mathcal{B}$$-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 $$x \geq y$$ 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 $$a \leq x \leq b$$ implies $$\lVert x \rVert \leq c \max\{\lVert a \rVert, \lVert b \rVert\}$$;
 * 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 $$0 \leq x \leq y$$ implies $$\lVert x \rVert \leq c \lVert y \rVert$$.

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 $$\mathcal{G}$$ is a family of bounded subsets of $$X,$$ then there are apparently no simple conditions guaranteeing that $$H$$ is a $$\mathcal{T}$$-cone in $$L_{\mathcal{G}}(X; Y),$$ even for the most common types of families $$\mathcal{T}$$ of bounded subsets of $$L_{\mathcal{G}}(X; Y)$$ (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 $$\left\{ X_{\alpha} : \alpha \in A \right\}$$ is a family of locally convex TVSs and that $$C_\alpha$$ is a cone in $$X_{\alpha}.$$ If $$X := \bigoplus_{\alpha} X_{\alpha}$$ is the locally convex direct sum then the cone $$C := \bigoplus_{\alpha} C_\alpha$$ 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 $$\mathcal{G}$$ is a family of bounded subsets of $$X,$$ then if the positive cone of $$X$$ is a $$\mathcal{G}$$-cone in $$X$$ and if the positive cone of $$Y$$ is a normal cone in $$Y$$ then the positive cone of $$L_{\mathcal{G}}(X; Y)$$ is a normal cone for the $$\mathcal{G}$$-topology on $$L(X; Y).$$