A new approach on Lipschitz compact operators

The notions of Lipschitz-free compact and Lipschitz-free weakly compact operators between metric spaces are introduced. Some nonlinear versions of Schauder's theorem and Gantmacher's theorem on compact and weakly compact linear operators are proved. The Davis–Figiel–Johnson–Pełczyński factorization theorem is stated for Lipschitz-free weakly compact operators.