Collectively compact operator approximation

Summary

A sequence of operators $K_n$ is a collectively compact approximation of an operator $K$ if $K_n \to K$ pointwise. In addition, the images of the unit ball under all $K_n$ together must form a relatively compact set. This weaker notion replaces norm convergence, which often fails for numerical methods such as the Nyström method. It still gives the main conclusion when $I - K$ is invertible. Then $I - K_n$ is invertible for large $n$, with uniformly bounded inverses, and the approximate solutions converge. The theory was developed by P. M. Anselone for the numerical solution of integral equations.