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In mathematics, specifically in order theory and functional analysis, if is a cone at the origin in a topological vector space such that and if is the neighborhood filter at the origin, then is called normal if where and where for any subset is the -saturatation of [1]

Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.

Characterizations

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

If is a collection of subsets of 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 then a cone in is called a -cone if is a fundamental subfamily of and is a strict -cone if is a fundamental subfamily of [1] Let denote the family of all bounded subsets of

If is a cone in a TVS (over the real or complex numbers), then the following are equivalent:[1]

  1. is a normal cone.
  2. For every filter in if then
  3. There exists a neighborhood base in such that implies

and if 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, -saturated sets.
  2. There exists a generating family of semi-norms on such that for all and

and if is a locally convex space and if the dual cone of 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 is the topology of uniform convergence on the equicontinuous subsets of

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

  1. there exists a Hausdorff locally compact topological space such that is isomorphic (as an ordered TVS) with a subspace of where is the space of all real-valued continuous functions on under the topology of compact convergence.[2]

If is a locally convex TVS, is a cone in with dual cone and is a saturated family of weakly bounded subsets of then[1]

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

If is a Banach space, is a closed cone in , and is the family of all bounded subsets of then the dual cone is normal in if and only if is a strict -cone.[1]

If is a Banach space and is a cone in then the following are equivalent:[1]

  1. is a -cone in ;
  2. ;
  3. is a strict -cone in

Ordered topological vector spaces

Suppose is an ordered topological vector space. That is, is a topological vector space, and we define whenever lies in the cone . The following statements are equivalent:[3]

  1. The cone is normal;
  2. The normed space admits an equivalent monotone norm;
  3. There exists a constant such that implies ;
  4. The full hull of the closed unit ball of is norm bounded;
  5. There is a constant such that implies .

Properties

Sufficient conditions

If the topology on 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 if and only if each is normal in [1]

If is a locally convex space then the closure of a normal cone is a normal cone.[1]

If is a cone in a locally convex TVS and if is the dual cone of then if and only if 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 and are ordered locally convex TVSs and if is a family of bounded subsets of then if the positive cone of is a -cone in and if the positive cone of is a normal cone in then the positive cone of is a normal cone for the -topology on [4]

See also

References

  1. ^ a b c d e f g h i j k l m n o p q r 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.
  4. ^ a b c Schaefer & Wolff 1999, pp. 225–229.

Bibliography