Normal cone (functional analysis)
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 

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.

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 
<li> 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:
<li>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

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:

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

Properties 

 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).

Sufficient conditions 

If the topology on  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  if and only if each  is normal in 

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

If  is a cone in a locally convex TVS  and if  is the dual cone of  then  if and only if  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  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

See also

References

Bibliography

  
  

Functional analysis