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]
- C is a normal cone.
- For every filter in X, if then .
- 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.
- 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 .
- 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 .
- is a -cone in .
- this means that the family is a fundamental subfamily of .
- 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:
- 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]
- if is a -cone then C is a normal cone for the -topology on X;
- 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]
- C is a -cone in X;
- ;
- 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]
References
- Schaefer & Wolff 1999, pp. 215–222.
- Schaefer & Wolff 1999, pp. 222-225.
- Schaefer & Wolff 1999, pp. 225–229.
- 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)