Fix $p \in \mathbb{N}$, given $(f_k)_k \subset L_p(\mathbb{R}^n)$. For every $T \in L_p(\mathbb{R}^n)^*$ (dual space), $(|T(f_k)|)_k< \infty$. Show that $(f_k)_k$ is also bounded in $L_p(\mathbb{R}^n)$.
I have to show that, $\int_X |f_k|^p dx <\infty$. Unfortunaly, I don't know how to start, so I am very thankful for hints.
Greetings.