I am trying to prove the following "contiuity-type" result.
Let $X,Y$ normed linear spaces. Let $\{T_n\} \to T \in \mathcal{L}(X,Y)$ and $\{u_n\} \to u \in X$. Show that $\{T_n(u_n)\} \to \{T(u)\} \in Y$
Any help will be most appreciated! Thank you very much in advance.