A priori inequality in proving $l^\infty$ is not $L^p(\mathbb T)$ multipliers

Question

Suppose for all $m\in l^\infty$, $T_mf=(m\hat f)^\vee$ is a bounded map on $L^p(\mathbb T)$, hence $\|T_mf\|_p\le C_m\|f\|_p$.

Then how can I use uniform boundness principle and get the priori inequality $\|T_mf\|_p\le C\|m\|_\infty\|f\|_p$? Since I can't find a good way to prove that $m\mapsto T_m$ is a continuous map.