Integral linear operator

In mathematical analysis, an integral linear operator is a linear operator T given by integration; i.e.,

(
        T
        f
        )
        (
        x
        )
        =
        ∫
        f
        (
        y
        )
        K
        (
        x
        ,
        y
        )
        
        d
        y
      
    
    {\displaystyle (Tf)(x)=\int f(y)K(x,y)\,dy}

where 
 
 
 
 K
 (
 x
 ,
 y
 )
 
 
 {\displaystyle K(x,y)}
 
 is called an integration kernel.
More generally, an integral bilinear form is a <a href="/facts/Bilinear_map/LH1lHVeP">bilinear functional</a> that belongs to the continuous dual space of 
 
 
 
 X
 
 
 
 
 ⊗
 ^
 
 
 
 
 ϵ
 
 
 Y
 
 
 {\displaystyle X{\widehat {\otimes }}_{\epsilon }Y}
 
, the <a href="/facts/Injective_tensor_product/53zV1QtL">injective tensor product</a> of the locally convex <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> (TVSs) X and Y. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form. 
These maps play an important role in the theory of <a href="/facts/Nuclear_space/4FLPWdfp">nuclear spaces</a> and <a href="/facts/Nuclear_map/kxnx5bIJ">nuclear maps</a>.

Integral linear operator open-in-new

Integral linear operator