"H=W" in infinite dimensions

It is well known that H^{m,p}(Ω) = W^{m,p}(Ω) holds for any m, n in N, p in [1, infty), and open subset Ω of 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 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.