On C¹ Whitney extension theorem in Banach spaces

Our note is a complement to recent articles {JS1} (2011) and {JS2} (2013) by M. Jim\'enez-Sevilla and L. S\'anchez-Gonz\'alez which generalise (the basic statement of) the classical Whitney extension theorem for C¹-smooth real functions on Rⁿ to the case of real functions on X ({JS1}) and to the case of mappings from X to Y ({JS2}) for some Banach spaces X and Y. Since the proof from {JS2} contains a serious flaw, we supply a different more transparent detailed proof under (probably) slightly stronger assumptions on X and Y. Our proof gives also extensions results from special sets (e.g. Lipschitz submanifolds or closed convex bodies) under substantially weaker assumptions on X and Y. Further, we observe that the mapping F∈ C¹(X;Y) which extends f given on a closed set A⊂ X can be, in some cases, C^∞-smooth (or Cᵏ-smooth with $k>1$) on X A. Of course, also this improved result is weaker than Whitney's result (for X= Rⁿ, Y= R) which asserts that F is even analytic on X A. Further, following another Whitney's article and using the above results, we prove results on extensions of C¹-smooth mappings from open (``weakly'') quasiconvex subsets of X. Following the above mentioned articles we also consider the question concerning the Lipschitz constant of F is f is a Lipschitz mapping. We also present a new observation on the limitation of the possible validity of Whitney's extension theorem for C²-smooth functions.