I am currently reading about operator topologies in the book "Methods of Modern Mathematical Physics: Functional Analysis" by Reed and Simon.
In their treatment of the Hilbert space adjoint, a theorem (Theorem VI.3) states that the adjoint map $T \mapsto T^*$ is always continuous in the weak and uniform topology, but is continuous in the strong operator topology only if the Hilbert space is finite-dimensional.
However, a previous comment (p.184) states that the weak operator topology is weaker than the strong operator topology which in turn is weaker than the uniform topology. Doesn't this mean that the continuity of the adjoint map in the uniform operator topology implies its continuity in the strong operator topology?
Unfortunately the proof for the theorem above is not really helpful for the beginner because it states that continuity in the uniform and weak topology are trivial. An accessible counterexample, however, is provided to demonstrate the case where continuity in the strong operator topology fails.
Thanks for helping me out of my confusion!