How to prove that the following norm, defined on $C([0,1],\mathbb{C})$ :
$||f||_p = (\int_0^1 |f(t)|^p dt)^{1/p}$ (for $p \in [1,+\infty[$) is "from" a scalar product only if $p=2$ ?
I don't know how to show it at all, I need food for thought... Thank you in advance !
Since your name is French, I assume with scalar product (produit scalaire) that you're talking about the inner product.
You could use that the parallelogram law holds if and only if the norm is induced by an inner product. So you only need to check that $$2\|f\|_p^2+2\|g\|_p^2=\|f+g\|_p^2+\|f-g\|_p^2 $$ holds for all $f,g\in C([0,1],\mathbb{C})$ only for $p=2$.
This post contains more details about the proof of the parallelogram law.