Normal cone (functional analysis)

In mathematics, specifically in order theory and functional analysis, if C is a cone at 0 in a topological vector space X such that 0 ∈ C and if ${\displaystyle {\mathcal {U}}}$ is the neighborhood filter at the origin, then 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 S of X, [S]_C := (S + C) ∩ (S − C) is the C-saturatation of S.[1]

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

If ${\displaystyle {\mathcal {T}}}$ is a collection of subsets of 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 X then a cone C in 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 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 X.

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

1. C is a normal cone.
2. For every filter ${\displaystyle {\mathcal {F}}}$ in 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 X such that ${\displaystyle B\in {\mathcal {G}}}$ implies ${\displaystyle \left[B\cap C\right]_{C}\subseteq B}$.

and if X is a vector space over the reals then we may add to this list:[1]

4. There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
5. There exists a generating family ${\displaystyle {\mathcal {P}}}$ of semi-norms on 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 X is a locally convex space and if the dual cone of 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 X is the topology of uniform convergence on the equicontinuous subsets of ${\displaystyle C^{\prime }}$.

and if 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 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 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.[2]

If X is a locally convex TVS, C is a cone in 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 C is a normal cone for the ${\displaystyle {\mathcal {G}}}$-topology on X;
2. if C is a normal cone for a ${\displaystyle {\mathcal {G}}}$-topology on 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 X is a Banach space, C is a closed cone in 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 C is a strict ${\displaystyle {\mathcal {B}}}$-cone.[1]

If X is a Banach space and C is a cone in X then the following are equivalent:[1]

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

Properties

• If X is a Hausdorff TVS then every normal cone in X is a proper cone.[1]
• If X is a normable space and if C is a normal cone in X then ${\displaystyle X^{\prime }=C^{\prime }-C^{\prime }}$.[1]
• 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 ${\displaystyle L_{s}\left(X;Y\right)}$ where H is the canonical positive cone of ${\displaystyle L\left(X;Y\right)}$ and ${\displaystyle L_{s}\left(X;Y\right)}$ is the space ${\displaystyle L\left(X;Y\right)}$ with the topology of simple convergence.[3]
  • If ${\displaystyle {\mathcal {G}}}$ is a family of bounded subsets of X, then there are apparently no simple conditions guaranteeing that H is a ${\displaystyle {\mathcal {T}}}$-cone in ${\displaystyle L_{\mathcal {G}}\left(X;Y\right)}$, even for the most common types of families ${\displaystyle {\mathcal {T}}}$ of bounded subsets of ${\displaystyle L_{\mathcal {G}}\left(X;Y\right)}$ (except for very special cases).[3]

Sufficient conditions

If the topology on 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 X if and only if each ${\displaystyle X_{\alpha }}$ is normal in ${\displaystyle X_{\alpha }}$.[1]

If X is a locally convex space then the closure of a normal cone is a normal cone.[1]

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

See also

• Cone-saturated
• Topological vector lattice
• Vector lattice

References

1. Schaefer & Wolff 1999, pp. 215–222.
2. Schaefer & Wolff 1999, pp. 222-225.
3. Schaefer & Wolff 1999, pp. 225–229.
4. Schaefer & Wolff 1999, pp. 225-229.

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)