Is the Hölder Space with the Hölder Norm Reflexive?

Question

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.

Answer 1

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.