Let $\mathbb E$ and $\mathbb F$ be two Banach spaces and $T:\mathbb E\to \mathbb F$ a linear operator. Use the definition of the weak topology to show that $T$ is continuous with respect $\mathbb E$ being equipped with the weak topology $\sigma(\mathbb E,\mathbb E^*)$ and $\mathbb F$ being equipped with the weak topology $\sigma(\mathbb F,\mathbb F^*)$ if and only if it is continuous at $0\in \mathbb E$ with respect to these weak topologies.
I think the idea behind is that continuity (weak) implies continuity (strong) and we know function is continuous iff it is continuous at one point. so this is also continuous(strong) at $0$, then again continuity(strong) implies continuity(weak) and we are done.
I am not sure whether my rough sketch is right or wrong? Also couldn't quite get the way of perfectly writing it. Any help would be highly appreciated. Thanks in advance.