How to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not Hilbert space? I want to show that  the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not a Hilbert space. So I should show that it is not an inner product space. Most likely, The Parallelogram Equality does not work for some two elements of this space but I could not find these two elements. Thanks for your helps.

Take $$\begin{cases}
f_1(x)=-1+2\frac{x-a}{b-a}\\
f_2(x)=\left\vert \frac{2x - (a+b)}{b-a} \right\vert
\end{cases}$$
You have $\Vert f_1 \Vert = \Vert f_2 \Vert=1$ and $\Vert f_1+f_2 \Vert = \Vert f_1-f_2 \Vert = 2$. Hence the parallelogram law is not satisfied.
