Let me clarify my question. Say $\{T_n\}$ is a sequence of bounded linear operators from $X$ to itself, where $X$ is a Banach Space. There exists a bounded linear operator $T$, s.t., $\lim_{n\rightarrow \infty}T_n(x)=T(x)\qquad\text{for every $x\in X$}.$
Now, under what additional condition will the following convergence hold, $\lim_{n\rightarrow \infty} ||T_n-T||=0?$