# 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?

• You want that $\left<f,f\right> = \Vert f \Vert^2$. Remember that $\vert f \vert^2 = ff^*$ with $f^*$ being the complex conjugate of $f$. Oct 18 '15 at 13:57
• You might want to look up "polarization identity." And then you might want to think about how lucky you are that your field doesn't have characteristic 2. Oct 18 '15 at 13:59
• Use polarization. Oct 18 '15 at 14:05

$$\langle f,g\rangle=\frac{1}{4}\left(||f+g||^2-||f-g||^2\right),$$ where $||f||$ is the norm that you described before.
As many has pointed out, what you need is the Polarization identity. Basically, it tells you when the converse of "every inner product induces a norm" is true. The special case where the field of the vector space is real is spelled out in Diego's post. For complex case the formula is $$\langle f,g \rangle=\frac 14 \sum_{n=0}^3 i^n||f+i^ng||^2$$ For more discussion and proofs, consider reading this link Norms Induced by Inner Products and the Parallelogram Law.