Let X=$C^{\alpha}(\Omega,\mathbb{R})$
be space of Hölder continuous functions. What is its dual?



Just a few words about how to get a representation for the dual, since the details on this topic are certainly treated in the literature. To fix notations, assume e.g. $\Omega\subset\mathbb{R}^n$ be an open neighborhood of $0$ , let $\Delta_\Omega\subset\Omega\times\Omega$ denote the diagonal, and $\tilde\Omega:=(\Omega\times\Omega)\setminus\Delta_\Omega\subset \mathbb{R}^{2n}$. Let $$\u\_\alpha:= u(0) + \sup _ {(x,y)\in\tilde\Omega}\frac{u(x)u(y)} {xy^\alpha}$$ be the usual $C^\alpha$ norm. We have therefore an isometric linear embedding $$j: C^ \alpha(\Omega) \to \mathbb{R}\times C^0_b(\tilde\Omega )$$ mapping $u\in C^ \alpha(\Omega)$ to the pair $\left( u(0), \frac{u(x)u(y)} {xy ^ \alpha} \right)$. This presents $C^ \alpha(\Omega)$ as a product of $\mathbb{R}$ and a subspace of the space of bounded continuous functions on the open set $\tilde\Omega$, the dual of which has a wellstudied representation. Lastly, recall that as a general fact, the dual of a product of two Banach spaces, endowed with the sumnorm, is the product of the duals, with their maxnorm; and that the dual of a subspace $Y$ of a Banach space $X$, is isometrically the quotient of the dual over the annichilator: $Y^* \sim X^*/Y^{\perp}$. edit. Of course if the uniqueness of representation is not relevant for you, you may skip the quotient. Thus, linear functionals $\phi$ on $C^0_b(\tilde\Omega)$, produce all continuous linear functionals on $C^\alpha(\Omega)$ via 


Your question can be interpreted in several ways, but I guess you are asking "is it a known space, or just some weird new Banach space?" If $\Omega=R^n$ and $s$ is noninteger, the dual of $C^s$ is known in the above sense (almost). Indeed, one can identify $C^s$ with the Besov space $B^s_{\infty,\infty}$, and duals of Besov spaces are well studied. The precise result is: denote with $\dot C^s$ the closure of the Schwartz space of rapidly decreasing functions in $C^s$, then $(\dot C^s)'=B_{1,1}^{s}$. I bet similar results should be true also on more general open sets $\Omega$ but I'm not sure. A good starting point are Triebel's books (Theory of Function Spaces I, II and III). 

