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.
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
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:
- is a normal cone.
- For every filter in if then
- 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:
- There exists a neighborhood base at the origin consisting of convex, balanced, -saturated sets.
- 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:
- For any equicontinuous subset there exists an equicontiuous such that
- 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:
- The topology of 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 is an ordered locally convex TVS over the reals whose positive cone is then we may add to this list:
- 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.
If is a locally convex TVS, is a cone in with dual cone and is a saturated family of weakly bounded subsets of then
- if is a -cone then is a normal cone for the -topology on ;
- 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.
If is a Banach space and is a cone in then the following are equivalent:
- is a -cone in ;
- ;
- is a strict -cone in
- If is a Hausdorff TVS then every normal cone in is a proper cone.
- If is a normable space and if is a normal cone in then
- Suppose that the positive cone of an ordered locally convex TVS is weakly normal in and that is an ordered locally convex TVS with positive cone If then is dense in where is the canonical positive cone of and is the space with the topology of simple convergence.
- If is a family of bounded subsets of then there are apparently no simple conditions guaranteeing that is a -cone in even for the most common types of families of bounded subsets of (except for very special cases).