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Normal cone

Main article: Normal cone (functional analysis)

If C is a cone in a TVS X then C is normal if U = [ U ] C {\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C}} , where U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin, [ U ] C = { [ U ] : U ∈ U } {\displaystyle \left[{\mathcal {U}}\right]_{C}=\left\{\left[U\right]:U\in {\mathcal {U}}\right\}} , and [ U ] C := ( U + C ) ∩ ( U − C ) {\displaystyle [U]_{C}:=\left(U+C\right)\cap \left(U-C\right)} is the C-saturated hull of a subset U of X.²

If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:³

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

and if X is a vector space over the reals then also:⁴

1. There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
2. There exists a generating family P {\displaystyle {\mathcal {P}}} of semi-norms on X such that p ( x ) ≤ p ( x + y ) {\displaystyle p(x)\leq p(x+y)} for all x , y ∈ C {\displaystyle x,y\in C} and p ∈ P {\displaystyle p\in {\mathcal {P}}} .

If the topology on X is locally convex then the closure of a normal cone is a normal cone.⁵

Properties

If C is a normal cone in X and B is a bounded subset of X then [ B ] C {\displaystyle \left[B\right]_{C}} is bounded; in particular, every interval [ a , b ] {\displaystyle [a,b]} is bounded.⁶ If X is Hausdorff then every normal cone in X is a proper cone.⁷

Properties

• Let X be an ordered vector space over the reals that is finite-dimensional. Then the order of X is Archimedean if and only if the positive cone of X is closed for the unique topology under which X is a Hausdorff TVS.⁸
• Let X be an ordered vector space over the reals with positive cone C. Then the following are equivalent:⁹

1. the order of X is regular.
2. C is sequentially closed for some Hausdorff locally convex TVS topology on X and X + {\displaystyle X^{+}} distinguishes points in X
3. the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.

See also

• Generalised metric – Metric geometry
• Order topology (functional analysis) – Topology of an ordered vector space
• Ordered field – Algebraic object with an ordered structure
• Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Ordered ring – ring with a compatible total orderPages displaying wikidata descriptions as a fallback
• Ordered vector space – Vector space with a partial order
• Partially ordered space – Partially ordered topological space
• Riesz space – Partially ordered vector space, ordered as a lattice
• Topological vector lattice

• 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.

References

1. Schaefer & Wolff 1999, pp. 222–225. - 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. https://search.worldcat.org/oclc/840278135 ↩

2. Schaefer & Wolff 1999, pp. 215–222. - 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. https://search.worldcat.org/oclc/840278135 ↩

3. Schaefer & Wolff 1999, pp. 215–222. - 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. https://search.worldcat.org/oclc/840278135 ↩

4. Schaefer & Wolff 1999, pp. 215–222. - 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. https://search.worldcat.org/oclc/840278135 ↩

5. Schaefer & Wolff 1999, pp. 215–222. - 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. https://search.worldcat.org/oclc/840278135 ↩

6. Schaefer & Wolff 1999, pp. 215–222. - 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. https://search.worldcat.org/oclc/840278135 ↩

7. Schaefer & Wolff 1999, pp. 215–222. - 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. https://search.worldcat.org/oclc/840278135 ↩

8. Schaefer & Wolff 1999, pp. 222–225. - 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. https://search.worldcat.org/oclc/840278135 ↩

9. Schaefer & Wolff 1999, pp. 222–225. - 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. https://search.worldcat.org/oclc/840278135 ↩