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

If $\mathcal$ is a collection of subsets of $X$ and if $\mathcal$ is a subset of $\mathcal$ then $\mathcal$ is a fundamental subfamily of $\mathcal$ if every $T \in \mathcal$ is contained as a subset of some element of $\mathcal.$
If $\mathcal$ is a family of subsets of a TVS $X$ then a cone $C$ in $X$ is called a $\mathcal$-cone if $\left\$ is a fundamental subfamily of $\mathcal$ and $C$ is a strict $\mathcal$-cone if $\left\$ is a fundamental subfamily of $\mathcal.$
Let $\mathcal$ 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 $\mathcal$ in $X,$ if $\lim \mathcal = 0$ then \lim \left \mathcal \rightC = 0.


3. There exists a neighborhood base $\mathcal$ in $X$ such that $B \in \mathcal$ implies \left B \cap C \rightC \subseteq B.
4. 
   

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

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

6. For any equicontinuous subset $S \subseteq X^,$ there exists an equicontiuous $B \subseteq C^$ such that $S \subseteq B - B.$


7. The topology of $X$ is the topology of uniform convergence on the equicontinuous subsets of $C^.$

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

8. The topology of $X$ is the topology of uniform convergence on strongly bounded subsets of $C^.$


9. $C^$ is a $\mathcal^$-cone in $X^.$
   * this means that the family $\left\$ is a fundamental subfamily of $\mathcal^.$
   


10. $C^$ is a strict $\mathcal^$-cone in $X^.$
    * this means that the family $\left\$ is a fundamental subfamily of $\mathcal^.$
    

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 $C^ \subseteq X^,$ and $\mathcal$ is a saturated family of weakly bounded subsets of $X^,$ then
# if $C^$ is a $\mathcal$-cone then $C$ is a normal cone for the $\mathcal$-topology on $X$;
# if $C$ is a normal cone for a $\mathcal$-topology on $X$ consistent with $\left\langle X, X^\right\rangle$ then $C^$ is a strict $\mathcal$-cone in $X^.$

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

If $X$ is a Banach space and $C$ is a cone in $X$ then the following are equivalent:
# $C$ is a $\mathcal$-cone in $X$;
# $X = \overline - \overline$;
# $\overline$ is a strict $\mathcal$-cone in $X.$

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 $X^ = C^ - C^.$
* 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 $L_(X; Y)$ is the space $L(X; Y)$ with the topology of simple convergence.
** If $\mathcal$ is a family of bounded subsets of $X,$ then there are apparently no simple conditions guaranteeing that $H$ is a $\mathcal$-cone in $L_(X; Y),$ even for the most common types of families $\mathcal$ of bounded subsets of $L_(X; Y)$ (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\$ is a family of locally convex TVSs and that $C_\alpha$ is a cone in $X_.$
If $X := \bigoplus_ X_$ is the locally convex direct sum then the cone $C := \bigoplus_ C_\alpha$ is a normal cone in $X$ if and only if each $X_$ is normal in $X_.$

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 $C^$ is the dual cone of $C,$ then $X^ = C^ - C^$ 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 $\mathcal$ is a family of bounded subsets of $X,$ then if the positive cone of $X$ is a $\mathcal$-cone in $X$ and if the positive cone of $Y$ is a normal cone in $Y$ then the positive cone of $L_(X; Y)$ is a normal cone for the $\mathcal$-topology on $L(X; Y).$

See also

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References

Bibliography

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{{Ordered topological vector spaces

Functional analysis