$L^2$ norm and $L^{\infty}$ inequality for periodic smooth functions
Remember that integrals are monotonic, so if $f\leq g$ a.e. then $\int f\leq \int g$. Thus if $g\geq 0$ a.e. and $f_1\leq f_2$ a.e. you get $f_1g\leq f_2g$ a.e. and thus $\int f_1g\leq \int f_2 g$.
So now see that $|\phi|\leq \|\phi\|_\infty$ a.e. and $(\phi f)\cdot(\phi f) = |\phi|^2 |f|^2$ and $|f|^2\geq 0$.
So using this we get $$ \|\phi f\|_2 \leq \Big\|\|\phi\|_\infty f\Big\|_2$$ and as $\|\phi\|_\infty$ is a scalar this is $$ \|\phi\|_\infty \|f\|_2$$