Continuity in the product of norm topology and weak* topology

Question

While I am reading the book on numerical range by Bonsall and Duncan, I encountered the following corollary. I got stuck in the following inequality which is about the continuity of the map $(x,f)\mapsto f(Tx)$. The symbols used here are the following:

$X$ is a Banach space and $\Pi:=\{(x,f)\in X\times X^*:~\|x\|=\|f\|=1, f(x)=1\}$ and $T$ is a bounded linear operator on $X$ and $V(T)$ is a set of scalars $V(T)=\{f(Tx):~(x,f)\in \Pi\}$. I do not understand how to tackle the 2nd quantity on the right hand side.

Thanks in advance.