# Strongly Continuous Multiplication on Bounded Subsets

In Pedersen's Analysis Now on page 171 there is an inequality which is given as an indication that multiplication is (jointly) strongly continuous on the subset $B(0,n) \times B(H) \subset B(H)$ (we only require the operators in the first coordinate to lie within some ball of $B(H)$). Suppose that $S_i \to S$ strongly and $T_i \to T$ strongly (these are nets). To see that $S_i T_i \to ST$ strongly we select a vector $x \in H$ and write $$||(ST-S_i T_i)x|| \leq ||(S-S_i)T_ix|| + ||S|| ||(T-T_i)x||.$$

We can make the right-hand summand as small as possible by letting $i \to \infty$. The left-hand summand is a little trickier, because the internal vector $T_i x$ varies with $i$.  How do I get the left-hand summand to become small?

-

I would expand the first term further, using $T_ix=Tx+(T_i-T)x$. Where boundedness of $\{S_i\}_i\cup\{S\}$ comes in is in showing that $\|(S-S_i)(T_i-T)x\|$ goes to zero.
$$\|(ST-S_iT_i)x\| \leq\|(S-S_i)Tx\|+\|S_i\|\|(T-T_i)x\|.$$