I have a question about tempered distributions. First, we know that every $f\in L^p(\mathbb{R}^n)\ (1\le p\le\infty)$ can be considered as a tempered distribution in the following sense: $$F_f:{\cal S}\ni\varphi\mapsto\int_{\mathbb{R}^n}f(x)\varphi(x)dx\in\mathbb{C}$$ where ${\cal S}$ denotes the set of all Schwartz functions. This procedure allows us to consider functions as tempered distributions. But can we do the inverse thing if some conditions are satisfied?
My thought is the following:
$F\in L^p$ for $F\in{\cal S}'$ if and only if there exists a function $f\in L^p$ such that $F=F_f$ in ${\cal S}'$.
And define $\|F\|_p=\|f\|_p$. However, this definition may depend on a function $f$ since the existence may not be unique. On the other hand, we can recall the following theorem:
Theorem 1. Let $1\le p\le\infty$ and let $f,g\in L^p$ be such that $F_f=F_g$ in ${\cal S}'$. Then, $f(x)=g(x)$ for a.e. $x$.
Thanks to it, the $L^p$-norm will be equal to each other whatever we choose a function $f$. In general, distributions are not functions, so if we want to consider them as functions, we need to define something.
Is my thought correct? Did I omit something important?