Let $m\in C[a,b]$ and consider $ T:(C[a,b],\Vert\cdot\Vert_1) \to (C[a,b],\Vert\cdot\Vert_1): f\mapsto m\cdot f $ Provided $\Vert T\Vert = \Vert m\Vert_\infty$, I want to show $\|T\|$ is attained iff there exi st $f\in C[a,b]\setminus\{0\}$ such that $\Vert m\Vert_\infty|f| = |m||f|$.
The 'if' direction is trivial, how to deal with the 'only if' part?
Thank you in advance.