文摘
Given a set-function \(\psi \) defined on bounded subsets of a Banach space with certain properties, necessary and sufficient criteria for \(\psi (A(U))=0\) are given, when A is positively homogeneous of some order and U is bounded. The results are applied to give necessary and sufficient criteria for the compactness and weak compactness of a Fréchet derivative (in some point or at \(\infty \)) and when an operator is improving.