In class, my professor said that for $\mu \in L^p(\mathbb{R}^n), p \in [1,\infty]$, we have a bounded linear operator $\mu':\mathcal{S(\mathbb{R}^n)} \rightarrow \mathbb{R}$ such that $\mu'(\phi):= \int \mu(x)\phi(x)dx$.
It is easy to see this is a linear operator, but how can I prove it's bounded? In particular, we'd then want to show $\exists M>0$ such that $\forall \phi \in \mathcal{S(\mathbb{R}^n)},$
$$ |\mu'(\phi)|= |\int \mu(x)\phi(x)dx| \leq M||\phi(x)||_{\mathcal{S(\mathbb{R}^n)}}$$ However, my question is:
a) How does $\mu \in L^p(\mathbb{R}^n), p \in [1,\infty]$ help here?
b) Since the Schwartz class comes with a countable collection of semi-norms, rather than a single norm, how can we make sense of $||\phi(x)||_{\mathcal{S(\mathbb{R}^n)}}$, and thus the bounding condition above?