Normal cone (functional analysis)

<p>In mathematics, specifically in <a href="/facts/Order_theory/gn6fM3oJ">order theory</a> and <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, if 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is a <a href="/facts/Convex_cone/h1tARfw4">cone</a> at the origin in a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 such that 
  
    
      
        0
        ∈
        C
      
    
    {\displaystyle 0\in C}
  
 and if 
  
    
      
        
          
            U
          
        
      
    
    {\displaystyle {\mathcal {U}}}
  
 is the <a href="/facts/Neighborhood_filter/fMRM8Xht">neighborhood filter</a> at the origin, then 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is called normal if 
  
    
      
        
          
            U
          
        
        =
        
          
            [
            
              
                U
              
            
            ]
          
          
            C
          
        
        ,
      
    
    {\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C},}
  
 where 
  
    
      
        
          
            [
            
              
                U
              
            
            ]
          
          
            C
          
        
        :=
        
          {
          
            [
            U
            
              ]
              
                C
              
            
            :
            U
            ∈
            
              
                U
              
            
          
          }
        
      
    
    {\displaystyle \left[{\mathcal {U}}\right]_{C}:=\left\{[U]_{C}:U\in {\mathcal {U}}\right\}}
  
 and where for any subset 
  
    
      
        S
        ⊆
        X
        ,
      
    
    {\displaystyle S\subseteq X,}
  
 
  
    
      
        [
        S
        
          ]
          
            C
          
        
        :=
        (
        S
        +
        C
        )
        ∩
        (
        S
        −
        C
        )
      
    
    {\displaystyle [S]_{C}:=(S+C)\cap (S-C)}
  
 is the <a href="/facts/Cone-saturated/sIaDXRLu">
  
    
      
        C
      
    
    {\displaystyle C}
  
-saturatation</a> of 
  
    
      
        S
        .
      
    
    {\displaystyle S.}

</p><p>Normal cones play an important role in the theory of <a href="/facts/Ordered_topological_vector_space/N1fvN8th">ordered topological vector spaces</a> and <a href="/facts/Topological_vector_lattice/q5BLo3cr">topological vector lattices</a>.
</p>

Normal cone (functional analysis) open-in-new

Normal cone (functional analysis)