An example of a not positive definite sesquilinear form

Question

Let $R\left(\mathbb{R}/\mathbb{Z}\right)$ denote the space of Riemann integrable periodic functions on $\left[0,1\right]$. Let $<\cdot,\cdot>: R\left(\mathbb{R}/\mathbb{Z}\right)\rightarrow\mathbb{R}$, such that $$<f,g>=\int_0^1f\left(x\right)\overline{g\left(x\right)} dx.$$

The fact that $<\cdot,\cdot>$ is not positive definite is given. I do not understand why. Can someone help me understanding why is this true?

Thanks!

Answer 1

Take $f$ to be the $1$-periodic function défined on $[0,1]$ by $f(x)=0$ if $0<x<1$, and $f(0)=f(1)=1$.

Then $f$ is nonzero, but $\langle f,f\rangle=0$.