Sobolev space reflexivity problem

Question

Let $ I $ open interval of $ \mathbb{R} $ We know that the Sobolev space $ W^{1, \infty}(I) $ is not reflexive. But, is there any easy proof of this result?

Thank you in advance

Answer 1

The space $W^{1,\infty}(I)$, $I = (a,b)$, is isomorphic to $\mathbb R \times L^\infty(I)$ via $$v \mapsto (v(a), v').$$ Since $L^\infty(I)$ is not reflexive, $W^{1,\infty}(I)$ cannot be reflexive.