Is $W^{k,p}$ a reflexiv Banach space?

Question

Let $U\subset \mathbb{R}^n$ be an open bounded subset with smooth boundary. Is the Banach space $W^{k,p}(U)$ a reflexiv Banach space? If not, for what $k$ and $p$ is it reflexive?

Answer 1

The Sobolev spaces, just like the $L^p$ spaces, are reflexive when $1 < p < \infty$. To show this, note that $W^{k,p}(U)$ embeds as a Banach space inside a finite product of $L^p(U)$ spaces as $$ u \mapsto (D^{\alpha}u)_{|\alpha| \leq k}. $$ A finite product of reflexive spaces is reflexive and a closed subspace of a reflexive space is reflexive and so $W^{k,p}(U)$ inherits it from the $L^p(U)$ spaces.