In my lecture, we are considering the vector space $$\{ f : \mathbb{R} \to \mathbb{R} \, | \, f(x+2\pi) = f(x) \, \forall x \in \mathbb{R} \text{ and } \| f \| < \infty \}$$ where $$\| f \| := \int_0^{2\pi} |f(t)|^2 dt$$
If we assume that $\| f \| < \infty$, then this means that $|f(t)|^2$ is Riemann-integrable. I asked myself whether I could say something about the Riemann integrability of $|f(t)|$ or even $f(t)$.
I know that the absolute value of a Riemann-integrable function is also Riemann-integrable. Furthermore, the square of a Riemann-integrable function is Riemann-integrable too. However, I am not sure whether the respective reverse statements are true or not.
Does anyone know something about it?