Let $n\in\mathbb N$ and $\Omega\subseteq\mathbb R^{n}$ be open. Consider the space $$H^{1}\left(\Omega\right):=\left\{f\in L^{2}\left(\Omega\right):\text{grad }f\in L^{2}\left(\Omega\right)^{n}\right\}$$ where $\text{grad}$ is the vector-valued gradient. I consider this space together with the norm $$\|g\|_{H_{1}\left(\Omega\right)}:=\left(\|g\|^{2}_{L_{2}\left(\Omega\right)}+\|\text{grad g}\|^{2}_{L_{2}\left(\Omega\right)^{n}}\right)^{\frac{1}{2}}$$ which can be identified as the corresponding graph-norm.
My problems: I am finding it hard to clearly understand what the space $H^{1}\left(\Omega\right)^{d}$ should look like, where $d\in\mathbb N$. Am I correct in my thinking that this space should be $$H^{1}\left(\Omega\right)^{d}=\left\{\phi=(\phi_{i})_{i\in\left\{1,\dots,d\right\}}\in L^{2}\left(\Omega\right)^{d}:\text{grad }\phi_{i}\in L^{2}\left(\Omega\right)^{n}\text{ for each }i\in\left\{1,\dots,d\right\}\right\}?$$ Am I correct in thinking that the corresponding norm in this case is $$\|\phi\|_{H^{1}\left(\Omega\right)^{d}}=\left(\|\phi\|^{2}_{L^{2}\left(\Omega\right)^{d}}+\sum_{i=1}^{d}\|\text{grad }\phi_{i}\|^{2}_{L^{2}\left(\Omega\right)^{n}}\right)^{\frac{1}{2}}?$$ If this is correct, is there some way to remove the sum in the sum of norms of the gradient parts, and present this more elegantly or concisely?