I was wondering if the following conditions are equivalent for a linear operator $T$ between $L^p$ spaces:
i) T maps $L^p$ to $L^p$, that is, if $f \in L^p$ then $Tf \in L^p$.
ii) There exists $C>0$ such that $\|Tf\|_{L^p} \leq C \| f \|_{L^p}$ for every $f \in L^p$.
It is clear to me that $ii)$ implies $i)$ but I can't see clearly the other implication. Can $T$ map $L^p$ to $L^p$ "unboundly"? Does it help if $T$ is bijective?
Thanks in advance.