Problem 5

Exercise 9.5 (Weakly converging operators)   Let $X$ and $Y$ be Banach spaces. A sequence $A_n\in L(X,Y)$ is said to converge weakly to $A\in L(X,Y)$ if for all $x\in X$ and all $\phi\in Y^*$, the sequence $\phi(A_n x)$ converges to $\phi(Ax)$. Assuming that $A_n$ converges weakly to $A$, show that $\sup_{n\geq 1}\Vert A_n\Vert<\infty$ and that the operator $A$ is bounded.

Proof. See Winter 2018 Exercise 6. $\qedsymbol$