Jaramillo Aguado, Jesús ÁngelJiménez Sevilla, María del MarSánchez González, L.2023-06-192023-06-192014-031088-682610.1090/S0002-9939-2013-11834-4https://hdl.handle.net/20.500.14352/33475In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete C-k Finsler manifold M is determined by the normed algebra C-b(k)(M) of all real-valued, bounded and C-k smooth functions with bounded derivative defined on M. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete C-k Finsler manifold M is determined by the algebra C-b(k)(M); (ii) the weak Finsler structure of a separable and complete C-k Finsler manifold M modeled on a Banach space with a Lipschitz and C-k smooth bump function is determined by the algebra C-b(k)(M); (iii) the weak Finsler structure of a C-1 uniformly bumpable and complete C-1 Finsler manifold M modeled on a Weakly Compactly Generated (WCG) Banach space is determined by the algebra C-b(1)(M); and (iv) the isometric structure of a WCG Banach space X with a C-1 smooth bump function is determined by the algebra C-b(1)(X).engjournal articlehttp://www.ams.org/journals/proc/2014-142-03/S0002-9939-2013-11834-4/http://www.ams.org/http://arxiv.org/pdf/1108.5403v1open access517Differentiable functionsRiemannian-manifoldsLipschitz functionsapproximationisometriesspacesAnálisis matemático1202 Análisis y Análisis Funcional