Does $\lim_{n\to \infty} ||A_n\varphi - A \varphi||_Y = 0$ imply $\lim_{n\to \infty} ||A_n|| = ||A||$ I am trying to understand different modes of convergence in a Banach space. Suppose we have a sequence of bounded linear operators $A_n:X\to Y$, where $X$ and $Y$ are Banach spaces, that converge pointwise and in operator norm to an operator $A$, i.e. we have \begin{align} \lim_{n\to \infty} ||A_n\varphi - A \varphi||_Y = 0, \\\ \lim_{n\to \infty} ||A_n - A || = 0. \end{align} Can we then say any/all/none of the following: 1. $\lim_{n\to \infty} A_n = A$ 2. $\lim_{n\to \infty} ||A_n|| = ||A||$ 3. $||\lim_{n\to \infty} A_n|| = ||A||$

Maybe this will helps you $$|\left\| {{A_n}} \right\| - \left\| A \right\|| \leqslant \left\| {{A_n} - A} \right\|$$ For the second question, the norme is a continuous application, so 2 and 3 are both correct.