Let $T:E\rightarrow F$ be a bounded linear map for E and F Banach spaces, and E reflexive. Let $B_E$ be the unitary closed ball in $E$. How would you argue that $T(B_E)$ is closed?
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If $E$ is reflexive, its closed unit ball is weakly compact, and $T$, being continuous from the weak topology to the weak topology, takes weakly compact sets to weakly compact sets.