I came across a lemma stated without proof and I couldn't figure out the proof so I would like to ask for help.
Suppose $f\in L^2$ is a sum of nonzero and disjointly supported $f_i$. Let $T$ be a bounded linear operator such that $Tf_i$'s are also disjointly supported. Then the lemma states that $$\frac{||Tf||}{||f||}\leq\sup_{i\in I}\frac{||Tf_i||}{||f_i||}.$$ In fact, I would like to have something similar for $L^p$ with $p\in[1,\infty)$.
This is obvious. Say $c$ is the sup on the right. Then $||Tf_j||_p\le c||f_j||_p$ for every $j$, so $$||Tf||_p^p=\sum||Tf_j||_p^p\le c^p\sum||f_j||_p^p=c^p||f||_p^p.$$