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 $S.$ ^[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 $S\subseteq X$ 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.^[1]

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 ${\mathcal {G}}.$ ^[1] 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:^[1]

$C$ 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:^[1]

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:^[1]

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 $C^{\prime }.$

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:^[1]

The topology of $X$ is the topology of uniform convergence on strongly bounded subsets of $C^{\prime }.$
$C^{\prime }$ $C^{\prime }$ is a ${\mathcal {B}}^{\prime }$ ${\mathcal {B}}^{\prime }$ -cone in $X^{\prime }.$ $X^{\prime }.$
- this means that the family $\left\{{\overline {\left[B^{\prime }\right]_{C}}}:B^{\prime }\in {\mathcal {B}}^{\prime }\right\}$ is a fundamental subfamily of ${\mathcal {B}}^{\prime }.$
$C^{\prime }$ $C^{\prime }$ is a strict ${\mathcal {B}}^{\prime }$ ${\mathcal {B}}^{\prime }$ -cone in $X^{\prime }.$ $X^{\prime }.$
- this means that the family $\left\{\left[B^{\prime }\right]_{C}:B^{\prime }\in {\mathcal {B}}^{\prime }\right\}$ is a fundamental subfamily of ${\mathcal {B}}^{\prime }.$

and if $X$ is an ordered locally convex TVS over the reals whose positive cone is $C,$ then we may add to this list:

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 $C^{\prime }\subseteq X^{\prime },$ 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 $X$ ;
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 $X_{b}^{\prime }$ then the dual cone $C^{\prime }$ is normal in $X_{b}^{\prime }$ if and only if $C$ is a strict ${\mathcal {B}}$ -cone.^[1]

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 $X$ ;
$X={\overline {C}}-{\overline {C}}$ ;
${\overline {C}}$ 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:^[3]

The cone $L_{+}$ is normal;
The normed space $L$ admits an equivalent monotone norm;
There exists a constant $c>0$ such that $a\leq x\leq b$ implies $\lVert x\rVert \leq c\max\{\lVert a\rVert ,\lVert b\rVert \}$ ;
The full hull $[U]=(U+L_{+})\cap (U-L_{+})$ of the closed unit ball $U$ of $L$ is norm bounded;
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.^[1]
If $X$ is a normable space and if $C$ is a normal cone in $X$ then $X^{\prime }=C^{\prime }-C^{\prime }.$ ^[1]
Suppose that the positive cone of an ordered locally convex TVS $X$ $X$ is weakly normal in $X$ $X$ and that $Y$ $Y$ is an ordered locally convex TVS with positive cone $D.$ $D.$ If $Y=D-D$ $Y=D-D$ then $H-H$ $H-H$ is dense in $L_{s}(X;Y)$ $L_{s}(X;Y)$ where $H$ $H$ is the canonical positive cone of $L(X;Y)$ $L(X;Y)$ and $L_{s}(X;Y)$ $L_{s}(X;Y)$ is the space $L(X;Y)$ $L(X;Y)$ with the topology of simple convergence.^[4]
- 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).^[4]

Sufficient conditions

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

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 $X_{\alpha }.$ ^[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 $C^{\prime }$ is the dual cone of $C,$ then $X^{\prime }=C^{\prime }-C^{\prime }$ 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 ${\mathcal {G}}$ 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 $L(X;Y).$ ^[4]

References

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r Schaefer & Wolff 1999, pp. 215–222.
^ Schaefer & Wolff 1999, pp. 222–225.
^ Aliprantis, Charalambos D. (2007). Cones and duality. Rabee Tourky. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-4146-4. OCLC 87808043.
^ ^a ^b ^c Schaefer & Wolff 1999, pp. 225–229.

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.

[FOOTNOTESchaeferWolff1999215–222-1] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r Schaefer & Wolff 1999, pp. 215–222.

[FOOTNOTESchaeferWolff1999222–225-2] Schaefer & Wolff 1999, pp. 222–225.

[3] Aliprantis, Charalambos D. (2007). Cones and duality. Rabee Tourky. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-4146-4. OCLC 87808043.

[FOOTNOTESchaeferWolff1999225–229-4] Schaefer & Wolff 1999, pp. 225–229.

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v t e Ordered topological vector spaces
Basic concepts	Ordered vector space Partially ordered space Riesz space Order topology Order unit Positive linear operator Topological vector lattice Vector lattice
Types of orders/spaces	AL-space AM-space Archimedean Banach lattice Fréchet lattice Locally convex vector lattice Normed lattice Order bound dual Order dual Order complete Regularly ordered
Types of elements/subsets	Band Cone-saturated Lattice disjoint Dual/Polar cone Normal cone Order complete Order summable Order unit Quasi-interior point Solid set Weak order unit
Topologies/Convergence	Order convergence Order topology
Operators	Positive State
Main results	Freudenthal spectral