On sequences of 𝐶^{𝑘,𝛿}_{𝑏} maps which converge in the uniform 𝐶⁰-norm

We study maps f ∈ C b k , δ ( U , Y ) f\in C^{k,\delta }_{b}(U,Y) and give detailed estimates on ‖ D k f ( x ) ‖ , x ∈ U , \|D^{k}f(x)\|,x\in U, in terms of ‖ f ‖ \|f\| and ‖ f ‖ k , δ \|f\|_{k,\delta } . These estimates are used to prove a lemma by D. Henry for the case k ≥ 2 k\geq 2 . Here U ⊂ X U\subset X is an open subset and X X and Y Y are Banach spaces.