Normal cone (functional analysis)

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]

See also

• Cone-saturated
• Topological vector lattice
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

References

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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

This page was last edited on 28 March 2023, at 06:02