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∈C and if 𝒰 is the neighborhood filter at the origin, then C is called normal if 𝒰=[𝒰]C, where [𝒰]C:={[U]C:U∈𝒰} and where for any subset S⊆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.

## Contents

* 1 Characterizations
* 1.1 Ordered topological vector spaces
* 2 Properties
* 3 Sufficient conditions
* 4 See also
* 5 References
* 6 Bibliography

## Characterizations

If C is a cone in a TVS X then for any subset S⊆X let [S]C:=(S+C)∩(S−C) be the C-saturated hull of S⊆X and for any collection 𝒮 of subsets of X let [𝒮]C:={[S]C:S∈𝒮}. If C is a cone in a TVS X then C is normal if 𝒰=[𝒰]C, where 𝒰 is the neighborhood filter at the origin.[1]

If 𝒯 is a collection of subsets of X and if ℱ is a subset of 𝒯 then ℱ is a fundamental subfamily of 𝒯 if every T∈𝒯 is contained as a subset of some element of ℱ. If 𝒢 is a family of subsets of a TVS X then a cone C in X is called a 𝒢-cone if {[G]C‾:G∈𝒢} is a fundamental subfamily of 𝒢 and C is a strict 𝒢-cone if {[G]C:G∈𝒢} is a fundamental subfamily of 𝒢.[1] Let 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 ℱ in X, if lim⁡ℱ=0 then lim⁡[ℱ]C=0.
3. There exists a neighborhood base 𝒢 in X such that B∈𝒢 implies [B∩C]C⊆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 𝒫 of semi-norms on X such that p(x)≤p(x+y) for all x,y∈C and p∈𝒫.

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:[1]

6. For any equicontinuous subset S⊆X′, there exists an equicontiuous B⊆C′ such that S⊆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 ℬ′ is the family of all strongly bounded subsets of X′ then we may add to this list:[1]

8. The topology of X is the topology of uniform convergence on strongly bounded subsets of C′.
9. C′ is a ℬ′-cone in X′.
* this means that the family {[B′]C‾:B′∈ℬ′} is a fundamental subfamily of ℬ′.
10. C′ is a strict ℬ′-cone in X′.
* this means that the family {[B′]C:B′∈ℬ′} is a fundamental subfamily of ℬ′.

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 C′⊆X′, and 𝒢 is a saturated family of weakly bounded subsets of X′, then[1]

1. if C′ is a 𝒢-cone then C is a normal cone for the 𝒢-topology on X;
2. if C is a normal cone for a 𝒢-topology on X consistent with ⟨X,X′⟩ then C′ is a strict 𝒢-cone in X′.

If X is a Banach space, C is a closed cone in X,, and ℬ′ is the family of all bounded subsets of Xb′ then the dual cone C′ is normal in Xb′ if and only if C is a strict 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 ℬ-cone in X;
2. X=C‾−C‾;
3. C‾ is a strict 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≥y whenever x−y lies in the cone L+. The following statements are equivalent:[3]

1. The cone L+ is normal;
2. The normed space L admits an equivalent monotone norm;
3. There exists a constant c>0 such that a≤x≤b implies ‖x‖≤cmax⁡{‖a‖,‖b‖};
4. The full hull [U]=(U+L+)∩(U−L+) of the closed unit ball U of L is norm bounded;
5. There is a constant c>0 such that 0≤x≤y implies ‖x‖≤c‖y‖.

## 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′=C′−C′.[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 Ls(X;Y) where H is the canonical positive cone of L(X;Y) and Ls(X;Y) is the space L(X;Y) with the topology of simple convergence.[4]
* If 𝒢 is a family of bounded subsets of X, then there are apparently no simple conditions guaranteeing that H is a 𝒯-cone in L𝒢(X;Y), even for the most common types of families 𝒯 of bounded subsets of L𝒢(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 {Xα:α∈A} is a family of locally convex TVSs and that Cα is a cone in Xα. If X:=⨁αXα is the locally convex direct sum then the cone C:=⨁αCα is a normal cone in X if and only if each Xα is normal in Xα.[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′ is the dual cone of C, then X′=C′−C′ 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 𝒢 is a family of bounded subsets of X, then if the positive cone of X is a 𝒢-cone in X and if the positive cone of Y is a normal cone in Y then the positive cone of L𝒢(X;Y) is a normal cone for the 𝒢-topology on L(X;Y).[4]

## See also

* Cone-saturated
* Topological vector lattice

## References

1. ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 Schaefer & Wolff 1999, pp. 215–222.
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. https://www.worldcat.org/oclc/87808043.
4. ↑ 4.0 4.1 4.2 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

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