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, C := ∩ is the C-saturatation of S.
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.
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.
Let denote the family of all bounded subsets of X.
If C is a cone in a TVS X, then the following are equivalent:

  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. 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. 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. 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 with a subspace of R, where R is the space of all real-valued continuous functions on X under the topology of compact convergence.
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. 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.
If X is a Banach space and C is a cone in X then the following are equivalent:
  1. C is a -cone in X;
  2. ;
  3. is a strict -cone in X.

    Properties

If the topology on X is locally convex then the closure of a normal cone is a normal cone.
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.
If X is a locally convex space then the closure of a normal cone is a normal cone.
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.
Every normal cone in a locally convex TVS is weakly normal.
In a normed space, a cone is normal if and only if it is weakly normal.
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.