I saw this space in the mathematical literature but I do not really understand what this means.
Is $H^k([a,b];H^m(\mathbb{R}^d))$ the same space as $H^{k,m}([a,b] \times \mathbb{R}^d)$?-Note that for $L^2$ spaces $L^2(A;L^2(B))= L^2(A \times B)$. So to work with this in a more explicit way, if I apply to some $u \in H^k([a,b];H^m(\mathbb{R}^d))$ the Laplacian to get $\Delta u$. Is this then an element of $H^k([a,b];H^{m-2}(\mathbb{R}^d))$?
Or another example question: Does the Schwartz-rule hold? So assume $k,m \ge 1$ is it then true that $\partial_t \partial_{x_2} u =\partial_t \partial_{x_2}u$ where $t \in [a,b]$ and $(x_1,...,x_n) \in \mathbb{R}^d$? Cause this would be true in $H^{k,m}([a,b] \times \mathbb{R}^d).$
If anything is unclear, please let me know.