Normal cone (functional analysis)

In mathematics, specifically in order theory and functional analysis, if C is a cone at 0 in a topological vector space X such that 0 ∈ C and if is the neighborhood filter at the origin, then C is called normal if , where and where for any subset S of 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.

Characterizations

If C is a cone in a TVS X then for any subset S of X let be the C-saturated hull of S of X and for any collection of subsets of X let . If C is a cone in a TVS X then C is normal if , where is the neighborhood filter at the origin.[1]

If is a collection of subsets of X and if is a subset of then is a fundamental subfamily of if every is contained as a subset of some element of . If is a family of subsets of a TVS X then a cone C in X is called a -cone if is a fundamental subfamily of and C is a strict -cone if is a fundamental subfamily of .[1] Let 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 in X, if then .
  3. There exists a neighborhood base in X such that implies .

and if X is a vector space over the reals then we may add to this list:[1]

  1. There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
  2. There exists a generating family of semi-norms on X such that for all and .

and if X is a locally convex space and if the dual cone of C is denoted by then we may add to this list:[1]

  1. For any equicontinuous subset , there exists an equicontiuous such that .
  2. The topology of X is the topology of uniform convergence on the equicontinuous subsets of .

and if X is an infrabarreled locally convex space and if is the family of all strongly bounded subsets of then we may add to this list:[1]

  1. The topology of X is the topology of uniform convergence on strongly bounded subsets of .
  2. is a -cone in .
    • this means that the family is a fundamental subfamily of .
  3. is a strict -cone in .
    • this means that the family is a fundamental subfamily of .

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

  1. 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 , and is a saturated family of weakly bounded subsets of , then[1]

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

If X is a Banach space, C is a closed cone in X,, and is the family of all bounded subsets of then the dual cone is normal in if and only if C is a strict -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 -cone in X;
  2. ;
  3. is a strict -cone in X.

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 .[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 where H is the canonical positive cone of and is the space with the topology of simple convergence.[3]
    • If is a family of bounded subsets of X, then there are apparently no simple conditions guaranteeing that H is a -cone in , even for the most common types of families of bounded subsets of (except for very special cases).[3]

Sufficient conditions

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

Suppose that is a family of locally convex TVSs and that is a cone in . If is the locally convex direct sum then the cone is a normal cone in X if and only if each is normal in .[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 is the dual cone of C, then 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 is a family of bounded subsets of X, then if the positive cone of X is a -cone in X and if the positive cone of Y is a normal cone in Y then the positive cone of is a normal cone for the -topology on .[4]

See also

References

  1. Schaefer & Wolff 1999, pp. 215–222.
  2. Schaefer & Wolff 1999, pp. 222-225.
  3. Schaefer & Wolff 1999, pp. 225–229.
  4. Schaefer & Wolff 1999, pp. 225-229.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
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