Jiménez Sevilla, María del MarSánchez González, Luis2023-06-202023-06-202011[[1] R. Aron and P. Berner, A Hahn-Banach extension theorem for analytic maps, Bull. Soc. Math. France 106 (1978), 3-24. [2] C.J. Atkin, Extension of smooth functions in infinite dimensions I: unions of convex sets, Studia Math. 146 (3) (2001), 201-226. [3] D. Azagra, R. Fry and A. Montesinos, Perturbed Smooth Lipschitz Extensions of Uniformly Continuous Functions on Banach Spaces, Proc. Amer. Math. Soc. 133 (2005), 727-734. [4] D. Azagra, J. Ferrera, F. López-Mesas, Y. Rangel Smooth approximation of Lipschitz functions on Riemannian manifolds, J. Math. Anal. Appl. 326 (2) (2007), 1370-1378. [5] D. Azagra, R. Fry and L. Keener, Smooth extension of functions on separable Banach spaces, Math. Ann. 347 (2) (2010), 285-297. [6] D. Azagra, R. Fry and L. Keener, Corrigendum to \Smooth extension of functions on separable Banach spaces", preprint. [7] R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics vol. 64, (1993). [8] M. Fabian, P. Habala, P. Hájek, V.M. Santalucía, J. Pelant and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Math. vol. 8, Springer-Verlag, New York, (2001). [9] R. Fry, Approximation by functions with bounded derivative on Banach spaces, Bull. Austr. Math. Soc. 69 (2004), 125-131. [10] P. Hájek and M. Johanis, Uniformly Gâteaux smooth approximation on c0(Γ), J. Math. Anal. Appl. 350 (2009), 623-629. [11] P. Hájek and M. Johanis, Smooth approximations, J. Funct. Anal. 259 (3) (2010), 561-582. [12] K. John, H. Torunczyk and V. Zizler, Uniformly smooth partitions of unity on superreflexive Banach spaces, Studia Math. 70 (1981), 129-137. [13] J.M. Lasry and P.L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math. 55 (3) (1986), 257-266. [14] J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 293-345. [15] N. Moulis, Approximation de fonctions differentiables sur certains espaces de Banach, Ann. Inst. Fourier (Grenoble) 21 (1971), 293-345. [16] M.E. Rudin, A new proof that metric spaces are paracompact, Proc. Amer. Math. Soc. 20 (2) (1969), 603.0022-247X10.1016/j.jmaa.2010.12.057https://hdl.handle.net/20.500.14352/41997Let us consider a Banach space X with the property that every real-valued Lipschitz function f can be uniformly approximated by a Lipschitz, C1-smooth function g with Lip(g)⩽CLip(f) (with C depending only on the space X). This is the case for a Banach space X bi-Lipschitz homeomorphic to a subset of c0(Γ), for some set Γ, such that the coordinate functions of the homeomorphism are C1-smooth (Hájek and Johanis, 2010 . Then, we prove that for every closed subspace Y⊂X and every C1-smooth (Lipschitz) function f:Y→R, there is a C1-smooth (Lipschitz, respectively) extension of f to X. We also study C1-smooth extensions of real-valued functions defined on closed subsets of X. These results extend those given in Azagra et al. (2010) [4] to the class of non-separable Banach spaces satisfying the above property.engjournal articlehttp://www.sciencedirect.com/science/journal/0022247Xopen access517.98Smooth extensionsSmooth approximationsAnálisis funcional y teoría de operadores