Hi I would like to know if there is a definition of Sobolev space as a set of mappings $f:U\subset \mathbb R^n \rightarrow \mathbb R^m$ (instead to $\mathbb R$). In particular what should be $H^{1,2}(U)_{\mathbb R^m}$. Does it make sense to define it as the set of maps $f=(f_1,...,f_m)$ such that $f_i \in H^{1,2}(U)$? What would be a natural norm? Would it be a Hilbert space as in the case where $f$ goes to $\mathbb R$?
thank you