Approximation property in terms of Lipschitz maps via tensor product approach

This article explores the extension of the classical approximation property and its variants to the nonlinear framework of Lipschitz operator theory. Building on Grothendieck's tensor product methodology, we characterize the Lipschitz approximation property of Banach spaces using Lipschitz finite-rank operators and tensor products. Furthermore, inspired by the $p$-approximation property defined via $p$-compact sets, we introduce and examine the Lipschitz $p$-approximation property. We also establish a factorization theorem for dual Lipschitz $p$-compact operators, mirroring known linear results. This paper looks more closely at how the Lipschitz approximation property and the $p$-approximation property of a Banach space are related to those of its Lipschitz-free space.