prove an operator is compact in reflexive space

Question

In Banach spaces $E,F$, a compact operator $T\in \mathcal{L}(E,F)$ maps weakly convergent sequences into strongly convergent sequences.

If E is reflexive, the converse is true. I need help in proving that as I don't know where to start. What property of reflexive spaces makes the operator convert bounded series in $E$ into strongly convergent subseries in $F$

Thank you for your help.

Answer 1

Use that in reflexive spaces, the closed unit ball is weakly sequentially compact. The equivalence of these two properties is the Eberlain-Smulyain theorem.

So all you use is the assumption that $\{T(x_n)\}$ is Cauchy if $\{x_n\}$ is weakly convergent, and apply the above fact to an arbitrary bounded sequence.