Auxiliary normed space

<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a branch of mathematics, two methods of constructing <a href="/facts/Normed_vector_space/rmDljfyE">normed spaces</a> from <a href="/facts/Absolutely_convex_set/rJ5HLWkg">disks</a> were systematically employed by <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> to define <a href="/facts/Nuclear_operator/kxnx5bIJ">nuclear operators</a> and <a href="/facts/Nuclear_space/4FLPWdfp">nuclear spaces</a>. 
One method is used if the disk 
  
    
      
        D
      
    
    {\displaystyle D}
  
 is bounded: in this case, the auxiliary normed space is 
  
    
      
        span
        ⁡
        D
      
    
    {\displaystyle \operatorname {span} D}
  
 with norm

p
          
            D
          
        
        (
        x
        )
        :=
        
          inf
          
            x
            ∈
            r
            D
            ,
            r
            >
            0
          
        
        r
        .
      
    
    {\displaystyle p_{D}(x):=\inf _{x\in rD,r>0}r.}
  
 
The other method is used if the disk 
  
    
      
        D
      
    
    {\displaystyle D}
  
 is <a href="/facts/Absorbing_set/GCtjsUuX">absorbing</a>: in this case, the auxiliary normed space is the <a href="/facts/Quotient_space_(linear_algebra)/WmQ3lqdI">quotient space</a> 
  
    
      
        X
        
          /
        
        
          p
          
            D
          
          
            −
            1
          
        
        (
        0
        )
        .
      
    
    {\displaystyle X/p_{D}^{-1}(0).}
  
 
If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as <a href="/facts/Topological_vector_spaces/XRex1ChN">topological vector spaces</a> and as <a href="/facts/Normed_space/rmDljfyE">normed spaces</a>).
</p>

Auxiliary normed space open-in-new

Auxiliary normed space