This is a problem from Brezis Exercise 3.11:
In the solution section, Brezis gave the following solution:
I wonder why does it suffices to argue on sequences and why does there exists some $y \in E$ such that $(Ax_n, y) \not\to (Ax, y)$? This might follow from the definition, but I am not seeing it.
It suffices to argue on sequences becase $E$ with the norm topology is a metric space; and in a metric space, continuity is equivalent to sequential continuity.
If $(Ax_n,y)\to (Ax,y)$ for all $y$, then $Ax_n\to Ax$ by defitition of weak$^*$-convergence. So if $Ax_n$ does not converge weak$^*$ to $Ax$, there has to exist $y$ such that $(Ax_n,y)\to (Ax,y)$ fails.
