If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that $$ \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}. $$ Similarly, we can define the Little Hölder space $\lambda_\alpha(X)$ as the subset of functions of $\Lambda_\alpha(X)$ satisfying that $$ \lim_{x \to y} \frac{|f(x) - f(y)|}{|x - y|^\alpha} = 0. $$ I recall a result stating that, at least in classical contexts like $X = \mathbb{T}^n$, $\Lambda_\alpha(X)$ is isomorphic to the double dual of $\lambda_\alpha(X)$.

Question:Is there any reference for that duality in the context of more general metric spaces? An initial google search didn't yield anything.