I have been working on the following problem.
Let $E$ and $F$ are two Banach spaces and $T\in \mathcal{L}(E,F)$ (the space of bounded linear operators from $E$ into $F$.) Suppose that $T$ is continuous from weak topology $\sigma(E,E^*)$ of $E$ to norm topology of $F$. Letting $F'=T(E)$, prove that $B_{F'}$ is compact.
Here, $B_{F'}$ is the closed unit ball in F'.
I have been trying to prove $B_{F'}$ is compact since then $T(E)$ is automatically finite dimensional by the consequence of Riesz lemma.
I even don't know if this is true or not. I just guess it should be true but I don't know why. I am keeping trying to find a convergent subsequence of any given sequence $(u_n)\subset B_{F'}$ but have not gotten anyting. Could anybody help me? Thanks in advance!
