Let $X,Y$ be normed spaces and $T:X\to Y$ be a linear map.
I need to show that $T$ is continuous $\iff$ for all $x_n\to 0$ in $X$ , $sup_n||Tx_n|| \lt \infty$.
If $T$ is continuous then if $x_n\to 0$ then $T(x_n) \to T(0)$ and in particular $sup ||T(x_n)|| $ is finite.
Im not sure how to do the other direction.
Thanks for helping.