I am reading Evans' PDE textbook and found to prove the existence of solutions to a PDE using a priori estimates, if one chooses the space $X=C([0,T];H^{s}(\mathbb R^d))$, then the norm is defined as $$\|u\|_{X}=\sup_{0\leq t\leq T} \|u\|_{H^s},$$ where $H^s$ is just Sobolev spaces.
My question is how to define the norm for the PDE if we choose $X=C^{1}([0,T];H^{s}(\mathbb R^d))$? Is it $$\|u\|_{X}=\sup_{0\leq t\leq T} \|u\|_{H^s}+\sup_{0\leq t\leq T} \|\partial_{t}u\|_{H^s}$$ or $$\|u\|_{X}=\sup_{0\leq t\leq T} \|u\|_{H^s}+\sup_{0\leq t\leq T} \|\nabla u\|_{H^s}?$$
The space $C^1([0,T],X)$ is the space of all functions from $[0,T]$ to $X$ that are continuously differentiable. The norm is the $\sup$-norm of the function and its time-derivative. So the first choice is correct. For the second one, there is the possibility of ambiguity as $\nabla$ is typically used to denote derivative with respect to spatial variables, not time.