Is $\mathcal C^1([a,b],\mathbb R)$ endowed with the $C^1$-norm a Hilbert space?

Question

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!

Answer 1

On $[a,b]=[0,1]$ set $f(x) = x$ and $g(x) = 1$. Then $$ \|f-g\|^2+\|f+g\|^2 = (\|x-1\|_\infty + 1)^2 + (\|x+1\|_\infty + 1)^2 = 4+9=13. $$ But $$ 2\|f\|^2+2\|g\|^2 = 2(1+1)^2 + 2 = 10. $$