Boundedness of sequence $ \{||T_n || \} $ in Banach space.

Question

I take a new course Funcional analiysis, and I need to prove:

Let $E$ and $F$ Banach spaces and $T_n (n = 1,2,...)$ linear continuous operators from $E$ in $F$. Suppose that for all $x \in E$ and for all $f \in F^*$ sequence of numbers $ \{f (T_n x) \}$ is bounded. Then sequence $ \{||T_n || \} $ is bounded. Where $ F^* $ denotes dual space. Could anyone help me?

Answer 1

This is a straight application of Banach Steinhauss.

https://en.wikipedia.org/wiki/Uniform_boundedness_principle