Let $f$ be a locally integrable function on $\mathbb{R}^n$. Prove that the functional $F$ on $\mathscr{D}$ defined by $\langle F, \phi \rangle = \int_{\mathbb{R}^n} f\phi$ is a distribution, where $\phi\in\mathscr{D}$ is a test function and $\mathscr{D}$ is he set of test functions.
I'm not at all sure where to even start with this one. Any help/hints would be very welcome.
A distribution is a continuous linear functional $F:\mathscr{D}\to\mathbb{R}$. $F$ is well-defined in our case because $f$ is locally integrable and each $\phi$ is compactly supported, and $F$ is linear since integration is linear. Thus it is enough to check continuity.
To do this, it is enough to show that for each compact set $K$ there is a real number $C\geq 0$ and a non-negative integer $N$ such that $$ |\langle F,\phi\rangle|\leq C\sum_{|\alpha|\leq N}\sup_K|\partial^{\alpha}\phi|$$ for all $\phi\in\mathscr{D}$ supported in $K$, where $|\alpha|=\alpha_1+\dots+\alpha_n$.
In this case, if $K$ is a compact set then $$ |\langle F,\phi\rangle|=\Big|\int_Kf\phi\Big|\leq \int_K|f||\phi|\leq ||f||_{L^1(K)}\sup |\phi| $$ so we can take $N=0$ and $C=||f||_{L^1(K)}$.