Convergence in $L^p$ implies convergence in $\mathcal{D}'(\Omega)$ Let $f_k, f \in L^p(\Omega)$, $k \in \mathbb{N}$ and $\Omega \subseteq \mathbb{R}^n$ open. The claim is:
$$ f_k \underset{k \rightarrow \infty}{\longrightarrow}f \text{ in } L^p(\Omega) \Rightarrow f_k \underset{k \rightarrow \infty}{\longrightarrow}f \text{ in } \mathcal{D}'(\Omega)$$
where $\mathcal{D}'(\Omega)$ is the set of distributions, i.e. all linear maps $F: \mathcal{D}(\Omega)= C_0^{\infty}( \Omega) \rightarrow \mathbb{R}$ which are continuous in the sense that if $\varphi_k \underset{k \rightarrow \infty}{\longrightarrow} \varphi \text{ in } \mathcal{D}(\Omega)$, we have $F(\varphi_k) \underset{k \rightarrow \infty}{\longrightarrow} F(\varphi)$.
I've read that the claim above should be fairly easy to show, however, I'm having a hard time doing this.
My thoughts so far: From $f_k \underset{k \rightarrow \infty}{\longrightarrow}f \text{ in } L^p(\Omega) $, we get that $|| f_k -f ||_{L^p(\Omega)} \leq \epsilon$ for some arbitrary $\epsilon >0$. Here, I thought that I would need to differentiate between $p< \infty$, giving $|| f_k -f ||_{L^p(\Omega)}= (\int_{\Omega} |f_k-f|^p dx)^{1/p}$, and $p= \infty$, giving $|| f_k -f ||_{L^p(\Omega)}= \underset{N \subseteq \Omega}{inf}~ \underset{ x \in \Omega \setminus N}{sup} |f_k-f|$, where $N$ is a set of measure zero. In order to show that $f_k \underset{k \rightarrow \infty}{\longrightarrow}f \text{ in } \mathcal{D}'(\Omega)$, I need to show that $f_k(\varphi) \underset{k \rightarrow \infty}{\longrightarrow}f( \varphi) ~\forall \varphi \in \mathcal{D}(\Omega)$ (1).
My problems are that, first of all, I'm not sure which norm I need to use in (1) (I assume the absolute norm) and how I should deduce (1) from what I know about $|| f_k -f ||_{L^p(\Omega)}$.
Thank you for your help and suggestions!

As you pointed out correctly we need to prove that
$$\int_\Omega f_k g\longrightarrow \int_\Omega f g$$
as $k\to\infty$ for every $g\in D(\Omega)$. This is equivalent to showing
$$\int_\Omega (f_k-f) g\longrightarrow 0$$
Now we can use Hölder's inequality as hinted by @User8128:
$$\left| \int_\Omega (f_k-f) g \right|\le \int_\Omega |(f_k-f) g|\le \|f_k-f\|_{L^p(\Omega)} \|g\|_{L^q(\Omega)}\to 0,$$
(where $\frac1p+\frac1q=1$) since $f_k\to f$ in $L^p(\Omega)$.
