Chain Rule for Conic Derivatives

For all “nice” definitions of differentiability, the Chain Rule should be valid. We show that the Chain Rule remains true for some wide class of definitions of differentiability if one considers as approximative mappings (derivatives) not just continuous linear, but positively homogeneous mappings satisfying certain topological conditions (which are fulfilled for continuous linear mappings). For brevity, we call such derivatives conic. We will give corollaries for the case of conic differentiation of mappings between normed spaces, especially for the case of Fréchet conic differentiation and compact conic differentiation.