In Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, he defines tempered distribution ($\mathscr S'$) as continuous linear functionals from the Schwartz class. Here, the continuity is given with respect to the family of seminorms \begin{equation*}\|\Phi\|_{\alpha,\beta}=\sup_{x\in\mathbb R^N}|x^\alpha\partial^\beta_x\Phi(x)|.\end{equation*} Then, without any further clarification, he proceeds to discuss convolutions between a tempered distribution and functions in the Schwartz family. This is where I get puzzled. If the tempered distributions are functionals, what is this convolution supposed to mean? It seems as if he (and every other source, for that matter) was assuming these functionals can be clearly identified with some functions. My question is: how do you make this identification?
I was thinking, as Schwartz functions are in $L^2$, maybe the identification is the one given by Riesz representation theorem. However, I think this is not possible as the topology we are considering in the Schwartz class is different from that of $L^2$. Moreover, while discussing $H^p$ spaces, he claims that, for $p>1$, $L^p$ is the same as $H^p$. Here, he is using again this identification that I don't quite get and, if my hypothesis was correct that the identification is made through Riesz representation theorem, this should mean $H^2=L^2=\mathscr S'$. This seems a bit strange to me. Another thing that's worrying me as well is the fact that he is discussing, without a prior definition, bounded distributions. Of course, if these were elements of $L^2$'s dual, they would be automatically bounded, so this is another hint that my original assumption about the identification is wrong.
I think this is a very basic question, but I don't find any source in which this is specifically discussed and clarify. How can we talk about an object as both a tempered distribution (an element of $\mathscr S'$) and a function defined on $\mathbb R^N$?