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# Hilbert-Schmidt integral operator

In mathematics, a Hilbert-Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert-Schmidt kernel is a function k : Ω × Ω → C with

$int_\left\{Omega\right\} int_\left\{Omega\right\} | k\left(x, y\right) |^\left\{2\right\} ,dx , dy < + infty$

and the associated Hilbert-Schmidt integral operator is the operator K : L2(Ω; C) → L2(Ω; C) given by

$\left(K u\right) \left(x\right) = int_\left\{Omega\right\} k\left(x, y\right) u\left(y\right) , dy.$

Hilbert-Schmidt integral operators are both continuous (and hence bounded) and compact.