Let $B:X\rightarrow Y$ be a bounded operator, I wish to show that $B:(X,wk)\rightarrow(Y,wk)$ is continuous.
So I started off by trying to show that $B$ is weakly continuous at 0. So we consider a weak neighborhood of $0_Y$ given by $W=\cap_{i=1}^n\{y\in Y:|\phi_i(y)|<\epsilon\}$ where $\phi_i\in Y^*$ and $\epsilon>0$. Our goal would then be to show that there exists a weak neighborhood of $0_X$ say $V$ such that $B(V)\subset W$. I can't see how to finish the proof... Is there a better approach?
Hint: $B^{-1}(W)=\bigcap_{i=1}^n\{x \in X,\,|(\phi_i \circ B)(x)| < \epsilon\}$.