# Reference request: $\alpha$-Hölder spaces as double duals

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.

First of all, I am not sure what you mean by $$L^\infty(X)$$ for a general complete metric space $$X$$. Don't you want $$C_b(X)$$?
Secondly: when $$X$$ is compact and $$0<\alpha<1$$, the result you want is Theorem 3.5 in
The result is proven in a more general setting, which includes noncompact $$X$$, in Theorem 4.38 of my book Lipschitz Algebras, second edition.