Continuous linear extension

<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, it is often convenient to define a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> on a <a href="/facts/Complete_space/lsckmU8G">complete</a>, <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 by first defining a linear transformation 
  
    
      
        L
      
    
    {\displaystyle L}
  
 on a <a href="/facts/Dense_set/nPT9Yxao">dense</a> <a href="/facts/Subset/SJGERmbA">subset</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and then <a href="/facts/Continuous_extension/sKbl02pB">continuously extending</a> 
  
    
      
        L
      
    
    {\displaystyle L}
  
 to the whole space via the theorem below. The resulting extension remains <a href="/facts/Linear_map/Q1PBKNVV">linear</a> and <a href="/facts/Bounded_operator/LYmDTIqR">bounded</a>, and is thus <a href="/facts/Continuous_function/sKbl02pB">continuous</a>, which makes it a continuous <a href="/facts/Linear_extension_(linear_algebra)/Q1PBKNVV">linear extension</a>. 
</p><p>This procedure is known as continuous linear extension.
</p>

Continuous linear extension open-in-new

Continuous linear extension