Sobolev spaces of vector-valued functions

Abstract: We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $\Omega\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(\Omega, V)$ with the so-called Sobolev-Reshetnyak space $R^{1,p}(\Omega, V)$. We see that, in general, $W^{1,p}(\Omega, V)$ is a closed subspace of $R^{1,p}(\Omega, V)$. As a main result, we obtain that $W^{1,p}(\Omega, V)=R^{1,p}(\Omega, V)$ if, and only if, the Banach space $V$ has the Radon-Nikod\'ym property