Let $X,Y$ be normed space and let $T_n \in B(X,Y)$ : Bounded Linear operators from $X$ to $Y$. I want to show that if
$\{f(T_n x)\}_n$ is bounded for any $x \in X$ and for any $f \in Y^*$,
then
$\{T_n x\}_n$ is also bounded for any $x \in X$.
How can I prove this?
This is the uniform boundedness principle. You have $$\sup_n \{ |(f\circ T_n)(x)|_\mathbb{K}\}<\infty$$ for all $f\in Y^*$ and all $x\in X$. So $$\sup_n\{ \|f \circ T_n\|_{X^*}\} <\infty $$ for all $f\in Y^*$. But this means $$\sup_{n}\{\|T_n^*(f)\|_{X^*}\}<\infty$$ for all $f\in Y^*$ and from the uniform boundedness principle again: $$\sup_n\{\|T^*_n\|_{B(Y^*,X^*)}\}<\infty$$ But that implies $\sup_n\{\|T_n(x)\|_{Y}\}<\infty.$