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In this work we consider natural generalizations of local complementation inBanach spaces, which include Lipschitz-local complementation. We show that allthese notions are indeed equivalent to the classical notion of localcomplementation of Banach spaces. As an application, we show that localcomplementation is naturally preserved under certain free constructions inFunctional Analysis, including Lipschitz-free spaces and free Banach lattices.

Local complementation in Banach spaces and its preservation under free constructions