document: A partial converse to Hadamard 's theorem on homeomorphisms A theorem by Hadamard gives a two-part condition under which a map from one Banach space to another is a homeomorphism . The theorem , while often very useful , is incomplete in the sense that it does not explicitly specify the family of maps for which the condition is met . Here , under a typically weak additional assumption on the map , we show that Hadamard 's condition is met if , and only if , the map is a homeomorphism with a Lipschitz continuous inverse . An application is given concerning the relation between the stability of a nonlinear system and the stability of related linear systems
doc_bio_tags: O B I O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: partial converse homeomorphisms banach space lipschitz continuous inverse
abstractive_keyphrases: hadamard theorem nonlinear system stability linear system stability linearization nonlinear feedback systems nonlinear networks