kernel, as above, is known as a Hilbert–Schmidt integral operator. Such operators are always compact. More generally, the Fredholm alternative is valid when K is any compact operator. The Fredholm alternative may be restated in the following form: a nonzero either is an eigenvalue of K, or lies in the domain of the resolvent Elliptic partial differential equations The Fredholm alternative can be applied to solving linear elliptic boundary value problems. The basic result is: if the equation and the appropriate Banach spaces have been set up correctly, then either (1) The homogeneous equation has a nontrivial solution, or (2) The inhomogeneous equation can be solved uniquely for each choice of data. The argument goes as follows. A typical simple-to-understand elliptic operator L would be the Laplacian plus some lower order terms. Combined with suitable boundary conditions and expressed on a suitable Banach space X (which encodes both the boundary conditions and the desired regularity of the solution), L becomes an unbounded operator from X to itself, and one attempts to solve where f ∈ X is some function serving as data for which we want a solution. The Fredholm alternative, together with the theory of elliptic equations, will enable us to organize the solutions of this equation. A concrete example would be an elliptic boundary-value problem like supplemented with the boundary condition where Ω ⊆ Rn is a bounded open set with smooth boundary and h(x) is a fixed coefficient function (a potential, in the case of a Schrödinger operator). The function f ∈ X is the variable data for which we wish to solve the equation. Here one would take X to be the space L2(Ω) of all square-integrable functions on Ω, and dom(L) is then the Sobolev space W 2,2(Ω) ∩ W(Ω), which amounts to the set of all square-integrable functions on Ω whose weak first and second derivatives exist and are square-integrable, and which satisfy a zero boundary condition on ∂Ω. If X has been selected correctly (as it has in this example), then for μ0 >> 0 the operator L + μ0 is positive, and then employing elliptic estimates, one can prove that L + μ0 : dom(L) → X is a bijection, and its inverse is a compact, everywhere-defined operator K from X to X, with image equal to dom(L). We fix one such μ0, but its value is not important as it is