Is the set of functions $f$ in $C^1([0,1])$ equipped with $\lVert f \lVert = \lVert f\lVert_\infty+\lVert f'\lVert_\infty$ (Case 1) or $\lVert f\lVert=\lVert f \lVert_\infty$ (Case 2) satisfying $\lvert f(0.5) \lvert \leq 1$ and $\int_0^1 \lvert f'(x)\lvert^2 dx \leq1$ relatively compact in $C^1([0,1])$?
I think this might be an application of the Arzela-Ascoli theorem. So I'd like to show that the set of functions is uniformly bounded and equicontinous. Boundedness is easy since
\begin{align}\lvert f(x)\lvert \leq 1+\int_{0.5}^x \lvert f'(t)\lvert dt\leq1+0.5+\int_0^1\lvert f'(t)\lvert^2 dt \quad \forall x. \end{align}
But equicontinuity seems harder since $\int_0^1 \lvert f'(x)\lvert^2 dx \leq1$ doesn't imply anything about the boundedness of $f'(x)$.