In chapter one of K-theory and $C^*$-algebras, a Friendly Approach, the author gives a very brief discussion about several types of continuity of operators between Hilbert spaces.
Let $T: \mathcal{H}_1\to\mathcal{H}_2$ be a linear operator between Hilbert spaces, we say $T$ is $\tau_1-\tau_2$ continuous if $T$ is continuous when $\mathcal{H}_1$ is equipped with topology $\tau_1$ and $\mathcal{H}_2$ is equipped with $\tau_2$.
The author says $T$ is norm-norm continuous iff $T$ is norm-weak continuous iff $T$ is weak-weak continuous, and that $T$ is weak-norm continuous iff $T$ is of finite rank.
I know that norm-norm countinuous implies norm-weak continuous, and weak-weak continuous implies norm-weak continuous. But I cannot figure out the rest of the assertion.
Can somebody give a hint? Also, before I continue to read the later parts of the book, I wonder whether these different types of continuity plays a crucial part in the theory.
Thanks!