Is the Banach space $ (C_0^1(0,T),\|\cdot\|_{C^1})$ separable?
We define $ C_0^1(a,b):=\{f\in C_0(a,b):\ f'\in C_0(a,b)\}, $ where $C_0(a,b)$ is the space of real-valued continuous functions on $[a,b]\subset \mathbb R$ such that $f(a)=f(b)=0$, $\|f\|_{C^1} :=\|f\|_\infty+\|f'\|_\infty$, where $\|\cdot\|_\infty$ is the supremum norm and $f'$ denotes the first derivative of $f$.