I'm studying the Fourier series on Zorich II book. In this context he introduces the vector space $\mathcal{R}_2([\mathbb{R},\mathbb{C}])$ of function that are "locally square-integrable, as proper or improper integral". He then says that in this space we can define an inner product as:
$$\left \langle f,g \right \rangle := \int_\mathbb{R} f(x)g^*(x)\mathrm{d}x$$
My doubt is: if this is an inner product, then it must be verified that $\left \langle f,f \right \rangle =0\iff f\underbrace{=}_{\text{in }\mathcal{R}_2([\mathbb{R},\mathbb{C}])}0$. So, who is the null "vector"? A vector in this context is a function, but there is not a unique function such that $\int_\mathbb{R} |f(x)|^2\mathrm{d}x = 0$ (it suffices for example that $f$ is continuous almost everywhere and is $\neq 0$ in its points of discontinuity).
If all the functions in the space $\mathcal{R}_2([\mathbb{R},\mathbb{C}])$ would be continuous, then that would be the case, but this is not.