I am referring to the introduction of this paper: G. Flores - On an identification of the Lipschitz-free spaces of $\mathbb{R}^n$
Here, the authors describe the predual space of the space
$$ Lip_0(M) := {\left\{ f: M \to \mathbb{R} ~ | ~ f(0_M)=0, ||f||_{Lip} < + \infty \right\} }$$
where $(M,d)$ is a "pointed" metric space (this just means that we fix a point $0_M$), and $||f||_{Lip} = { \sup_{x \neq y} \frac{|f(x)-f(y)|}{||x-y||}}$ is the Lipschitz norm.
Then, they define the space
$$ \mathcal{F}(M) = \overline{ \text{span}\left\{ ev_x | x\in M \right\} } $$
where $ev_x \in Lip_0(M)^*$ is the evaluation function at a point $x$, i.e. $ev_x(f) = f(x)$, and they state that
$$ \mathcal{F}(M)^* = Lip_0(M) $$
My question is: it is always true that we can embed a space in its double dual, therefore
- $ \mathcal{F}(M)^* = Lip_0(M) \subset (Lip_0(M)^*)^* $
Moreover, by definition, $\mathcal{F}(M) \subset Lip_0(M)^*$, from which follows immediately that
- $ \mathcal{F}(M)^* \supset (Lip_0(M)^*)^* $
But then I would conclude that $Lip_0(M) = Lip_0(M)^{**}$, i.e. $Lip_0(M)$ would be reflexive, which is not stated anywhere and seems too good to be true.
Can anyone spot the mistake in this reasoning, if there is any?
