Let $X,Y$ be normed linear spaces (or Banach spaces if necessary) and let $T: X \to Y$ be linear. We call $T$ norm-norm continuous if $X,Y$ are endowed with the norm topology and similarly, weak-weak continuous if $X,Y$ are endowed with the weak topology.
I am trying to show that if $T$ is norm norm continuous then it is weak-weak continuous. My idea was to use the sequential definition of continuity and to show that if $x_n \to x$ weakly then $Tx_n \to Tx$ weakly. That was easy enough but to complete my proof I would now have to show that this implies that $T$ is continuous and I can't seem to prove it. It would be easy if the topologies were the norm topologies but with both spaces carrying the weak topology I don't see how to proceed.
My question is: Is it true that if $T$ is linear and $x_n \to x$ weakly implies $Tx_n \to Tx$ weakly then $T$ is continuous? If yes, could someone please show me a proof, I can't seem to work it out.