Show that norm is induced by a scalar product Consider $I = [-1,1]$. Let $C(I)$ be the normed space, equipped with norm
\begin{align}
||f||_{2} = \left( \int_{-1}^{1} |f(t)|^2 \, dt \right) ^{1/2}
\end{align}
Show, that norm is induced by a scalar product. Any hints on how to proceed? 

Any norm in a Hilbert space is induced by a scalar product, to do this you can use the polarization identity:
$$\langle f,g\rangle=\frac{1}{4}\left(||f+g||^2-||f-g||^2\right),$$
where $||f||$ is the norm that you described before.
