I don't really understand the difference between weak convergence and strong convergence of operators.
According to my script, strong convergence of a bounded operators $A_n,A$ on a Hilbert space $H$ means:
$A_n x \rightarrow Ax$ for all $x \in H$.
And weak convergence means:
$<y,A_n x> \ \rightarrow \ <y,A x>$ for all $x,y \in H$.
Strong convergence clearly implies weak convergence. But i don't understand why weak convergence doesn't imply strong convergence.
If we have
$\lim_{n \rightarrow \infty}<y,A_n x>=<y,Ax>$,
we can pull the limit inside the scalar product to obtain:
$<x,\lim_{n \rightarrow \infty} A_n y>=<y,Ax>$.
Since this is true for all $x,y \in H$ we obtain strong convergence. Where is the mistake in here?