不眠不休考虑了几天，只好到这投医啦。
The statement is:
Suppose T is a linear operator from two Banachs X->Y.
Given any sequence (in X) x_n -> 0 and any bounded linear functional defined on Y, denoted by f, we have that f(T(x_n)) ->0. Now I want to use this fact to induce that T is continuous at 0。

The statement above further implies everywhere continuity as T is a linear operator. So the continuity at 0
is crucial to prove. I tried to use Hahn-Banach extension theorem, but did not go much far.