Let $(X,d)$ be an uncountable infinite complete disconnected metric space (what I have in mind is something like $X=\{0,1,\ldots,n\}^{\mathbb{N}}$). I would like to know if the space $C^{\gamma}(X)$ of all continuous real functions satisfying $\mathrm{Hol}(f) = sup\_{x\neq y} (|f(x)-f(y)|)/d^{\gamma}(x,y)<+\infty$ endowed with the norm  

$\| f\|\_{\gamma} = \|f\|\_{\infty} + \mathrm{Hol}(f)$ 
is a isomorphic to a reflexive Banach space ? 


I strongly suspect that the answer is no. I contacted some specialists regarding this question but no one was able to give me an answer. I shall add that the complete disconnected hypothesis could not help very much, but I just added it because of the classical argument based on Banach-Alaoglu and Krein-Milman preventing reflexivity. Unless I am not seeing something obvious it seems that is not trivial to find a isomorphic copy of $\ell\_1$ on this space. Any help is appreciated. 



Is the Hölder Space with the Hölder Norm Reflexive?
I suppose you are assuming $X$ is compact? Because $\|f\|\_\infty$ need not be finite in general.


Anyway, for any infinite metric space $X$ the space ${\rm Lip}(X)$ is not isomorphic to a reflexive Banach space. This includes Holder spaces as a special case ($f$ is $\alpha$-Holder for the metric $d$ iff it is Lipschitz for the metric $d^\alpha$). I use the norm ${\rm max}(\|f\|\_\infty, L(f))$ (for very good reasons), but it is isomorphic to the norm you mention so it doesn't change the question.


If $X$ is compact let $(p\_n)$ be a sequence of distinct points which converges to some point not in the sequence. If $X$ is not compact, let $(p\_n)$ be a uniformly discrete sequence. In either case, for each $n$ we have $c\_n = \inf\_m\{d(p\_n,p\_m)\} > 0$. Set $d\_n = {\rm min}(\frac{c\_n}{2}, 1)$ and let $\tau\_n$ be the function $$\tau\_n(q) = (d\_n - d(p\_n,q)) \vee 0.$$ Then define a map from $l^\infty$ into ${\rm Lip}(X)$ by taking the sequence $(a\_n)$ to the function $\sum a\_n \tau\_n$. This isomorphically embeds $l^\infty$ in ${\rm Lip}(X)$ (indeed, isometrically if for each $n$ there exists $q \neq p\_n$ with $d(p\_n,q) \leq d\_n$), which shows that ${\rm Lip}(X)$ is not isomorphic to a reflexive Banach space.


