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



> 
> W. G. Bade, P. C. Curtis, Jr, and H. G. Dales,
>  *Amenabilty and weak amenability for Beurling and Lipschitz algebras.*
>  Proc. London Math. Soc. 55 (1987) 359–377.
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