Why a linear operator is bounded if $|f(x)|\leqslant C\|x\|$ It's written in my book that $L:(X,|\cdot |_X)\to (Y,|\cdot |_y)$ is bounded if there is $C>0$ s.t. $$|L(x)|_Y\leqslant C|x|_X.$$
This definition seems very strange to me. Where is the motivation behind this definition ?

The motivation: A linear operator $L$ is bounded if $L$ is bounded on the closed unit ball in $X$.
