Normal cone (functional analysis)

In mathematics, specifically in order theory and functional analysis, if [math]\displaystyle{ C }[/math] is a cone at the origin in a topological vector space [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ 0 \in C }[/math] and if [math]\displaystyle{ \mathcal{U} }[/math] is the neighborhood filter at the origin, then [math]\displaystyle{ C }[/math] is called normal if [math]\displaystyle{ \mathcal{U} = \left[ \mathcal{U} \right]_C, }[/math] where [math]\displaystyle{ \left[ \mathcal{U} \right]_C := \left\{ [ U ]_C : U \in \mathcal{U} \right\} }[/math] and where for any subset [math]\displaystyle{ S \subseteq X, }[/math] [math]\displaystyle{ [S]_C := (S + C) \cap (S - C) }[/math] is the [math]\displaystyle{ C }[/math]-saturatation of [math]\displaystyle{ S. }[/math]^[1]

Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.

Characterizations

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

If [math]\displaystyle{ \mathcal{T} }[/math] is a collection of subsets of [math]\displaystyle{ X }[/math] and if [math]\displaystyle{ \mathcal{F} }[/math] is a subset of [math]\displaystyle{ \mathcal{T} }[/math] then [math]\displaystyle{ \mathcal{F} }[/math] is a fundamental subfamily of [math]\displaystyle{ \mathcal{T} }[/math] if every [math]\displaystyle{ T \in \mathcal{T} }[/math] is contained as a subset of some element of [math]\displaystyle{ \mathcal{F}. }[/math] If [math]\displaystyle{ \mathcal{G} }[/math] is a family of subsets of a TVS [math]\displaystyle{ X }[/math] then a cone [math]\displaystyle{ C }[/math] in [math]\displaystyle{ X }[/math] is called a [math]\displaystyle{ \mathcal{G} }[/math]-cone if [math]\displaystyle{ \left\{ \overline{\left[ G \right]_C} : G \in \mathcal{G} \right\} }[/math] is a fundamental subfamily of [math]\displaystyle{ \mathcal{G} }[/math] and [math]\displaystyle{ C }[/math] is a strict [math]\displaystyle{ \mathcal{G} }[/math]-cone if [math]\displaystyle{ \left\{ \left[ G \right]_C : G \in \mathcal{G} \right\} }[/math] is a fundamental subfamily of [math]\displaystyle{ \mathcal{G}. }[/math]^[1] Let [math]\displaystyle{ \mathcal{B} }[/math] denote the family of all bounded subsets of [math]\displaystyle{ X. }[/math]

If [math]\displaystyle{ C }[/math] is a cone in a TVS [math]\displaystyle{ X }[/math] (over the real or complex numbers), then the following are equivalent:^[1]

1. [math]\displaystyle{ C }[/math] is a normal cone.
2. For every filter [math]\displaystyle{ \mathcal{F} }[/math] in [math]\displaystyle{ X, }[/math] if [math]\displaystyle{ \lim \mathcal{F} = 0 }[/math] then [math]\displaystyle{ \lim \left[ \mathcal{F} \right]_C = 0. }[/math]
3. There exists a neighborhood base [math]\displaystyle{ \mathcal{G} }[/math] in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ B \in \mathcal{G} }[/math] implies [math]\displaystyle{ \left[ B \cap C \right]_C \subseteq B. }[/math]

and if [math]\displaystyle{ X }[/math] 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, [math]\displaystyle{ C }[/math]-saturated sets.
5. There exists a generating family [math]\displaystyle{ \mathcal{P} }[/math] of semi-norms on [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ p(x) \leq p(x + y) }[/math] for all [math]\displaystyle{ x, y \in C }[/math] and [math]\displaystyle{ p \in \mathcal{P}. }[/math]

and if [math]\displaystyle{ X }[/math] is a locally convex space and if the dual cone of [math]\displaystyle{ C }[/math] is denoted by [math]\displaystyle{ X^{\prime} }[/math] then we may add to this list:^[1]

6. For any equicontinuous subset [math]\displaystyle{ S \subseteq X^{\prime}, }[/math] there exists an equicontiuous [math]\displaystyle{ B \subseteq C^{\prime} }[/math] such that [math]\displaystyle{ S \subseteq B - B. }[/math]
7. The topology of [math]\displaystyle{ X }[/math] is the topology of uniform convergence on the equicontinuous subsets of [math]\displaystyle{ C^{\prime}. }[/math]

and if [math]\displaystyle{ X }[/math] is an infrabarreled locally convex space and if [math]\displaystyle{ \mathcal{B}^{\prime} }[/math] is the family of all strongly bounded subsets of [math]\displaystyle{ X^{\prime} }[/math] then we may add to this list:^[1]

8. The topology of [math]\displaystyle{ X }[/math] is the topology of uniform convergence on strongly bounded subsets of [math]\displaystyle{ C^{\prime}. }[/math]
9. [math]\displaystyle{ C^{\prime} }[/math] is a [math]\displaystyle{ \mathcal{B}^{\prime} }[/math]-cone in [math]\displaystyle{ X^{\prime}. }[/math]
   • this means that the family [math]\displaystyle{ \left\{ \overline{\left[ B^{\prime} \right]_C} : B^{\prime} \in \mathcal{B}^{\prime} \right\} }[/math] is a fundamental subfamily of [math]\displaystyle{ \mathcal{B}^{\prime}. }[/math]
10. [math]\displaystyle{ C^{\prime} }[/math] is a strict [math]\displaystyle{ \mathcal{B}^{\prime} }[/math]-cone in [math]\displaystyle{ X^{\prime}. }[/math]
    • this means that the family [math]\displaystyle{ \left\{ \left[ B^{\prime} \right]_C : B^{\prime} \in \mathcal{B}^{\prime} \right\} }[/math] is a fundamental subfamily of [math]\displaystyle{ \mathcal{B}^{\prime}. }[/math]

and if [math]\displaystyle{ X }[/math] is an ordered locally convex TVS over the reals whose positive cone is [math]\displaystyle{ C, }[/math] then we may add to this list:

11. there exists a Hausdorff locally compact topological space [math]\displaystyle{ S }[/math] such that [math]\displaystyle{ X }[/math] is isomorphic (as an ordered TVS) with a subspace of [math]\displaystyle{ R(S), }[/math] where [math]\displaystyle{ R(S) }[/math] is the space of all real-valued continuous functions on [math]\displaystyle{ X }[/math] under the topology of compact convergence.^[2]

If [math]\displaystyle{ X }[/math] is a locally convex TVS, [math]\displaystyle{ C }[/math] is a cone in [math]\displaystyle{ X }[/math] with dual cone [math]\displaystyle{ C^{\prime} \subseteq X^{\prime}, }[/math] and [math]\displaystyle{ \mathcal{G} }[/math] is a saturated family of weakly bounded subsets of [math]\displaystyle{ X^{\prime}, }[/math] then^[1]

1. if [math]\displaystyle{ C^{\prime} }[/math] is a [math]\displaystyle{ \mathcal{G} }[/math]-cone then [math]\displaystyle{ C }[/math] is a normal cone for the [math]\displaystyle{ \mathcal{G} }[/math]-topology on [math]\displaystyle{ X }[/math];
2. if [math]\displaystyle{ C }[/math] is a normal cone for a [math]\displaystyle{ \mathcal{G} }[/math]-topology on [math]\displaystyle{ X }[/math] consistent with [math]\displaystyle{ \left\langle X, X^{\prime}\right\rangle }[/math] then [math]\displaystyle{ C^{\prime} }[/math] is a strict [math]\displaystyle{ \mathcal{G} }[/math]-cone in [math]\displaystyle{ X^{\prime}. }[/math]

If [math]\displaystyle{ X }[/math] is a Banach space, [math]\displaystyle{ C }[/math] is a closed cone in [math]\displaystyle{ X, }[/math], and [math]\displaystyle{ \mathcal{B}^{\prime} }[/math] is the family of all bounded subsets of [math]\displaystyle{ X^{\prime}_b }[/math] then the dual cone [math]\displaystyle{ C^{\prime} }[/math] is normal in [math]\displaystyle{ X^{\prime}_b }[/math] if and only if [math]\displaystyle{ C }[/math] is a strict [math]\displaystyle{ \mathcal{B} }[/math]-cone.^[1]

If [math]\displaystyle{ X }[/math] is a Banach space and [math]\displaystyle{ C }[/math] is a cone in [math]\displaystyle{ X }[/math] then the following are equivalent:^[1]

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

Ordered topological vector spaces

Suppose [math]\displaystyle{ L }[/math] is an ordered topological vector space. That is, [math]\displaystyle{ L }[/math] is a topological vector space, and we define [math]\displaystyle{ x \geq y }[/math] whenever [math]\displaystyle{ x - y }[/math] lies in the cone [math]\displaystyle{ L_+ }[/math]. The following statements are equivalent:^[3]

1. The cone [math]\displaystyle{ L_+ }[/math] is normal;
2. The normed space [math]\displaystyle{ L }[/math] admits an equivalent monotone norm;
3. There exists a constant [math]\displaystyle{ c \gt 0 }[/math] such that [math]\displaystyle{ a \leq x \leq b }[/math] implies [math]\displaystyle{ \lVert x \rVert \leq c \max\{\lVert a \rVert, \lVert b \rVert\} }[/math];
4. The full hull [math]\displaystyle{ [U] = (U + L_+) \cap (U - L_+) }[/math] of the closed unit ball [math]\displaystyle{ U }[/math] of [math]\displaystyle{ L }[/math] is norm bounded;
5. There is a constant [math]\displaystyle{ c \gt 0 }[/math] such that [math]\displaystyle{ 0 \leq x \leq y }[/math] implies [math]\displaystyle{ \lVert x \rVert \leq c \lVert y \rVert }[/math].

Properties

• If [math]\displaystyle{ X }[/math] is a Hausdorff TVS then every normal cone in [math]\displaystyle{ X }[/math] is a proper cone.^[1]
• If [math]\displaystyle{ X }[/math] is a normable space and if [math]\displaystyle{ C }[/math] is a normal cone in [math]\displaystyle{ X }[/math] then [math]\displaystyle{ X^{\prime} = C^{\prime} - C^{\prime}. }[/math]^[1]
• Suppose that the positive cone of an ordered locally convex TVS [math]\displaystyle{ X }[/math] is weakly normal in [math]\displaystyle{ X }[/math] and that [math]\displaystyle{ Y }[/math] is an ordered locally convex TVS with positive cone [math]\displaystyle{ D. }[/math] If [math]\displaystyle{ Y = D - D }[/math] then [math]\displaystyle{ H - H }[/math] is dense in [math]\displaystyle{ L_s(X; Y) }[/math] where [math]\displaystyle{ H }[/math] is the canonical positive cone of [math]\displaystyle{ L(X; Y) }[/math] and [math]\displaystyle{ L_{s}(X; Y) }[/math] is the space [math]\displaystyle{ L(X; Y) }[/math] with the topology of simple convergence.^[4]
  • If [math]\displaystyle{ \mathcal{G} }[/math] is a family of bounded subsets of [math]\displaystyle{ X, }[/math] then there are apparently no simple conditions guaranteeing that [math]\displaystyle{ H }[/math] is a [math]\displaystyle{ \mathcal{T} }[/math]-cone in [math]\displaystyle{ L_{\mathcal{G}}(X; Y), }[/math] even for the most common types of families [math]\displaystyle{ \mathcal{T} }[/math] of bounded subsets of [math]\displaystyle{ L_{\mathcal{G}}(X; Y) }[/math] (except for very special cases).^[4]

Sufficient conditions

If the topology on [math]\displaystyle{ X }[/math] is locally convex then the closure of a normal cone is a normal cone.^[1]

Suppose that [math]\displaystyle{ \left\{ X_{\alpha} : \alpha \in A \right\} }[/math] is a family of locally convex TVSs and that [math]\displaystyle{ C_\alpha }[/math] is a cone in [math]\displaystyle{ X_{\alpha}. }[/math] If [math]\displaystyle{ X := \bigoplus_{\alpha} X_{\alpha} }[/math] is the locally convex direct sum then the cone [math]\displaystyle{ C := \bigoplus_{\alpha} C_\alpha }[/math] is a normal cone in [math]\displaystyle{ X }[/math] if and only if each [math]\displaystyle{ X_{\alpha} }[/math] is normal in [math]\displaystyle{ X_{\alpha}. }[/math]^[1]

If [math]\displaystyle{ X }[/math] is a locally convex space then the closure of a normal cone is a normal cone.^[1]

If [math]\displaystyle{ C }[/math] is a cone in a locally convex TVS [math]\displaystyle{ X }[/math] and if [math]\displaystyle{ C^{\prime} }[/math] is the dual cone of [math]\displaystyle{ C, }[/math] then [math]\displaystyle{ X^{\prime} = C^{\prime} - C^{\prime} }[/math] if and only if [math]\displaystyle{ C }[/math] 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 [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are ordered locally convex TVSs and if [math]\displaystyle{ \mathcal{G} }[/math] is a family of bounded subsets of [math]\displaystyle{ X, }[/math] then if the positive cone of [math]\displaystyle{ X }[/math] is a [math]\displaystyle{ \mathcal{G} }[/math]-cone in [math]\displaystyle{ X }[/math] and if the positive cone of [math]\displaystyle{ Y }[/math] is a normal cone in [math]\displaystyle{ Y }[/math] then the positive cone of [math]\displaystyle{ L_{\mathcal{G}}(X; Y) }[/math] is a normal cone for the [math]\displaystyle{ \mathcal{G} }[/math]-topology on [math]\displaystyle{ L(X; Y). }[/math]^[5]

See also

• Cone-saturated
• Topological vector lattice

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

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