In Brezis' Functional Analysis, exercise 6.2, item 1 is the following
Let $E$ and $F$ two Banach spaces, and $T: E \rightarrow F$ a continuous linear operator.
Assume $E$ is reflexive. Prove that $T(B_E)$ is closed (strongly).
In pg 62 ones read
suppose that T is continuous from $E$ weak into $F$ weak. Then $G(T)$ is closed in $E \times F$ equipped with the product topology $\sigma(E, E^*) \times \sigma(F, F^*)$ (...). It fallows that $G(T)$ is strongly closed.
Theorem 3.10 gives the above continuity for $T$ in weak topology and with Kakutani's Theorem ensures that $B_E$ is compact in the weak topology. The above comment is enough to finish the proof for the question?
Thanks in advance