In C*-Algebras and Operator Theory by Murphy, it is stated in Chapter 4.2 that on a norm-bounded closed set, the ultra-weak topology and (relative) weak topology coincide. However, it later asks in an exercise to prove that a weakly convergent sequence of operators is necessarily norm-bounded. Doesn't this imply that any weakly convergent sequence is ultra-weakly convergent?
Namely, if $(u_k)$ is a weakly convergent sequence(to say $u$), then it is contained a closed ball of radius $r$. But then on this closed ball, $(u_k)$ is ultra-weakly convergent to the same limit $u$.
But there is clearly something wrong here as ultra-weak topology and weak topology do not necessarily coincide but I am not sure why. I would like to see where it goes wrong in this reasoning as we have an apparent contradiction.
You’re right that any weakly convergent sequence is ultraweakly convergent to the same limit. But this does not hold if you replace sequence by net. In infinite dimensions neither of these topologies is second countable, so you can’t just use convergence of sequences to characterize these topologies.