Let $\mathcal C^1([a,b],\mathbb R)$ be endowed with the $C^1$-norm $\|\cdot\|$ where $\|f\| = \| f \|_\infty + \| f' \|_\infty$ for $f \in \mathcal C^1([a,b],\mathbb R)$.
In class, we have proved that $\langle \mathcal C^1([a,b],\mathbb R), \|\cdot\| \rangle$ is a Banach space. I could not figure out if $\langle \mathcal C^1([a,b],\mathbb R), \|\cdot\| \rangle$ is a Hilbert space or not. In other words, I'm not sure if $\|\cdot\|$ is induced by some inner product.
Could you please shed me some light on this issue? Thank you for your help!