The given question is
Let $X,U$ be Banach spaces and suppose that $U$ is reflexive, let $M:X\rightarrow U$ be a bounded linear map and suppose that $x_n$ converges weakly to $x$ in $X$ denoted $x_n\rightarrow_w x$. Show that $Mx_n\rightarrow_w Mx$ in $U$.
What I'm wondering is if the fact that $U$ is reflexive is superfluous information in this case. Clearly if $l$ is a bounded linear functional over $U$ then $l\circ M$ is a bounded linear functional over $X$ and therefore $l(Mx_n)\rightarrow l(Mx)$. Am I missing something here? What could be the authors idea with demanding $U$ to be reflexive?
