Consider $\Omega \subset \mathbb R^m$ an open. Define the Sobolev space $$W^{1, 2}(\Omega; \mathbb R^n):=\{ g \in L^2(\Omega; \mathbb R^n) \ : \ D g \in L^2(\Omega; \mathbb R^{n \times m}) \},$$ where $Dg$ is the distributional Jacobian of $g$ (i.e. the derivatives are meant in the distributional sense). $W^{1, 2}(\Omega; \mathbb R^n)$ is Hilbert with $$ \langle h, g \rangle_{W^{1, 2}(\Omega;\mathbb R^n)}:=\ \langle h,g \rangle_{L^2(\Omega; \mathbb R^n)} \ +\ \langle Dh,Dg \rangle_{L^2(\Omega; \mathbb R^{m \times n})}. $$ Here by $$\langle Dh,Dg \rangle_{L^2(\Omega; \mathbb R^{m \times n})}$$ I mean the product component by component of the matrixes $Dh$ and $Dg$.
Is everything correct?