Let $T \in \mathcal{L}(E,F)$ where $E, F$ are two normed vector spaces (not necessarily finite or complete (Banach)). Is it true that if $x_{n} \rightarrow x$ in $E$, is it true that $Tx_{n} \rightarrow y \in F$?
I know that it is true in finite dimensions that $Tx_{n} \rightarrow Tx$, and that we have a similar result for weakly converging sequences, so I'm assuming that there is a counterexample. I just can't find one.