Given an open and bounded subset $\,\Omega\subset\mathbb{R}^n\,$ and the interval $\,[0,T]\,$ such that $\,T<\infty$, I'm trying to find a norm for $\,X=\mathcal{C}^1\left([0,T]\times\Omega\right)$ such that $X$ is a Banach space.
I know the norm for one variable functions is $$\|f\|_{\,\mathcal{C}^1\left([0,T]\right)}=\|f(t)\|_\infty+\|f'(t)\|_\infty$$ so I tried this for $\,u=u(x,t),\;(x,t)\in\Omega\times[0,T]$ $$\|u\|_{X}=\max_{0\leq t\leq T}\|u(\cdot,t)\|_{\infty}+\max_{0\leq t\leq T}\|\,|Du|(\cdot,t)\,\|_{\infty}$$ where $\,|\cdot|\,$ denotes the modulus.
But honestly, I'm not sure this is correct. Thanks in advance for the help.