Determine whether $\langle f, g \rangle = \int_{-1}^1 f'(x) g'(x) \,dx$ is an inner product on $C^1[-1, 1]$

Question

Let $S = C^1[-1,1]$ functions, and define $$\langle f , g \rangle = \int_{-1}^{1} f'(x)g'(x) \,dx .$$ Decide whether $\langle \,\cdot\, , \,\cdot\,\rangle$ is an inner product on $S$.

To decide whether this is an inner product I'm going through the axioms, and I can't show this one:

$\langle f,f \rangle \geq 0$, and equality holds if and only if $f=0$ .

I know that this inner product fails in this axiom for that particular space of functions, but I don't know how to tackle this. Any ideas?

Answer 1

Hint We will have $\langle f , g \rangle = \int_{-1}^1 f' g' dx = 0$ if, for example $f' = 0$, that is, if $f$ is constant.