Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite-dimensional Haar subspace of C(X,Rk), the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,Rk) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C(X,Rk).