I am reading Brezis's book and I have trouble understanding a proof. I put up an image of the relevant part of the text.
What I don't understand is towards the end when he writes:
"It follows from (4) that $\varphi$ is continuous at $0$ for the topology $\sigma(E^*,E)$..."
I understand why $W$ is a neighborhood of $0$ but I don't understand how the continuity of $\varphi$ follows from (4).