It is well-known that if $U\subset \mathbb{R}^n$ is a open subset, $k\in\{1,2, \dots\}$ and $0<\lambda\le 1$ then $C^{k,\lambda}(\bar{U})$ (endowed with the sup norm) is a Banach space.
Is this true in a more general setting? For instance, given an open set $U\subset \mathbb{G}$, where $\mathbb{G}$ is a Carnot group, is the space $C^{k,\lambda}_{\mathbb{G}}(\overline{U})$ endowed with the norm
a Banach space? Is there any reference about this? Does it follow trivially from the Euclidean case?
I recall that, given a multi-index $I$, we denote higher order derivatives by $X^I=X_1^{i_1}\dots X_N^{i_N}$ and $D^{I}=\left( \frac{\partial}{\partial x} \right)^{I}=\frac{\partial^{i_{1}}}{\partial x_{1}^{i_{1}}}\dots \frac{\partial^{i_{N}}}{\partial x_{1}^{i_{N}}}$.