It is well known that $H^{ m,p}(\Omega) = W^{ m,p}(\Omega)$ holds for any $m, n \in \mathbb{ N}$, $p \in [1, \infty)$, and open subset $\Omega$ of $\mathbb{ R}^n$. Due to the essential difficulty that there exists no nontrivial translation-invariant measure in infinite dimensions, it is hard to obtain its infinite-dimensional counterparts. In this paper, using infinite-dimensional analogues of the classical techniques of truncation, boundary straightening, and partition of unity, we prove that smooth cylinder functions are dense in $W^{ m,p}(O)$. Consequently, $H^{ m,p}(O) = W^{ m,p}(O)$ holds for any $m \in \mathbb{ N}$, $p \in [1, \infty)$, and open subset $O$ of $\ell^2$ with a suitable boundary. Moreover, in the key step of compact truncation, we also prove that the Schatten $p$-norm type estimates for the higher-order derivatives of the Gross convolution are sharp for $p = 2$.
Categories: math.FA math.AP