Bounded operator and image of convergent net is compact implies compact operator

Question

Let $X$ and $Y$ be Banach spaces with $X$ reflexive. I now have a bounded operator $T \colon X \to Y$ that also has the property that if $(x_i)_{i \in I}$ is a bounded net that weakly converges, then the net $(Tx_i)_{i \in I}$ is convergent in norm. I now have to prove that $T$ is a compact but I have no idea on how to start.