AbstractThe paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: LetGbe a bounded domain, β be a tensor‐valued function onGsatisfying certain restrictions, and let {n} be a sequence of vector‐valued functions onGwhere theL2‐norms of {n}, {curln}, and {div(βn)} are bounded, and where allneither satisfy xn= 0 or (βFn) = 0 at the boundary ∂GofG( = normal to ∂G): then {n} has aL2‐convergent subsequence. The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ∂Gif β is interpreted as electric dielectricity ϵ or as magnetic permeability μ, respectively.These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for