Projective tensor product

<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, an area of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the projective tensor product of two <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex topological vector spaces</a> is a natural topological vector space structure on their <a href="/facts/Tensor_product/7K3pR1ap">tensor product</a>. Namely, given locally convex topological vector spaces 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
, the projective topology, or π-topology, on 
  
    
      
        X
        ⊗
        Y
      
    
    {\displaystyle X\otimes Y}
  
 is the <a href="/facts/Final_topology/gk2UV5va">strongest</a> topology which makes 
  
    
      
        X
        ⊗
        Y
      
    
    {\displaystyle X\otimes Y}
  
 a locally convex topological vector space such that the canonical map 
  
    
      
        (
        x
        ,
        y
        )
        ↦
        x
        ⊗
        y
      
    
    {\displaystyle (x,y)\mapsto x\otimes y}
  
 (from 
  
    
      
        X
        ×
        Y
      
    
    {\displaystyle X\times Y}
  
 to 
  
    
      
        X
        ⊗
        Y
      
    
    {\displaystyle X\otimes Y}
  
) is continuous. When equipped with this topology, 
  
    
      
        X
        ⊗
        Y
      
    
    {\displaystyle X\otimes Y}
  
 is denoted 
  
    
      
        X
        
          ⊗
          
            π
          
        
        Y
      
    
    {\displaystyle X\otimes _{\pi }Y}
  
 and called the projective tensor product of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
. It is a particular instance of a <a href="/facts/Topological_tensor_product/eDICDJ2Y">topological tensor product</a>.
</p>

Projective tensor product open-in-new

Projective tensor product