I'm learning the distribution theory about the principal value and the question arises as I read about the statement that says $H(f)=\frac{1}{\pi}pv \left( \frac{1}{x} \right) \ast f$, where $f$ is in Schwartz space.
By definition, $$\frac{1}{\pi}pv \left( \frac{1}{x} \right) \ast f=\lim_{\epsilon\to 0}\frac 1\pi\int_{\mid y\mid\geq \epsilon} f(x-y)\frac{dy}y.$$ But it is the fact that the right hand side converges to $H(f)$ when we take the limit in the $L^2$ sense.
The book does not discuss this and I was wondering if we define the principal value in such a way ($L^2$ convergence). Thanks for your help!