Bounded linear operator in subset implies bounded in the whole space?

Question

Let $T$ be an operator from $L^p$ to $L^p$, $S$ a dense subset of $L^p$, such that for any $f \in S$, $||T(f)|| < C||f||$.

Then, can we affirm this property extends to all $f$ in $L^p$. If not, please give a counterexample or additional conditions.