In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain Ω in n -dimensional Euclidean space Rn , then the square-integrable function k : Ω × Ω → C belonging to L2(Ω×Ω) such that is called a Hilbert–Schmidt kernel and the associated integral operator T : L2(Ω) → L2(Ω) given by is called a Hilbert–Schmidt integral operator. Then T is a Hilbert–Schmidt operator with Hilbert–Schmidt norm Hilbert–Schmidt integral operators are both continuous and compact. The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let L2(X) be a separable Hilbert space and X a locally compact Hausdorff space equipped with a positive Borel measure. The initial condition on the kernel k on Ω ⊆ Rn can be reinterpreted as demanding k belong to L2(X × X) . Then the operator is compact. If then T is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces.