# Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial.

The proof begins with defining a norm on the space of test functions $D(\mathbb{R}^d)$ by $$\|\psi\|=\int_{T^d}\int_{\mathbb{R}^d}|\hat{\psi}(t+r\omega)|dm_d(t)d\sigma_d(\omega),$$where $r>0$ is fixed, $T^d$ is the torus in $\mathbb{C}^d$, $\sigma_d$ the Haar measure for this torus, and $m_d$ is the Lebesgue measure for $\mathbb{R}^d$.

Then the proof goes on well, and this norm is used to prove the existence of a distribution, that is, our fundamental solution.

However, I do not know how Rudin thinks about this strange norm. It reminds of the norm on Sobolev spaces but you do not have the integration over torus there.