Let $A_{n}$ be a sequence of bounded linear operators defined on the Hilbert space ($A_{n}\in \mathcal{B}(\mathcal{H})$) such that $A_{n}$ strongly converges to $A\in \mathcal{B}(\mathcal{H})$. Additionally, assume $C$ to be a compact operator defined on the same space. Prove that $A_{n}C$ converges uniformly to $AC$.
It is possibly an easy question but since I’m new to this topic I couldn’t have it solved yet. Should I somehow use the fact that the image of a bounded set by $C$ is a pre-compact set? What good tool will this act provide me to obtain uniform convergence?
Thank you in advance.
If $C=0$ then this is trivially true so assume $C \neq 0$. Define the compact set $D := \overline{C(B_1[0])} $ then as $T_n \rightarrow T$ pointwise on $D$ then $T_n \rightarrow T$ uniformly on $D$ (see this link for a proof of how this works). As $T_n \rightarrow T$ uniformly on $D$ then $T_nC \rightarrow T$ uniformly as required.