Integral linear operator

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

${\displaystyle (Tf)(x)=\int f(y)K(x,y)\,dy}$

where ${\displaystyle K(x,y)}$ is called an integration kernel.

More generally, an integral bilinear form is a bilinear functional that belongs to the continuous dual space of ${\displaystyle X{\widehat {\otimes }}_{\epsilon }Y}$, the injective tensor product of the locally convex topological vector spaces (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 nuclear spaces and nuclear maps.