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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 $i$. [Edit] How do I get the left-hand summand to become small?

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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.

Come to think of it, you'd have one less step if you instead used the inequality

$$\|(ST-S_iT_i)x\| \leq\|(S-S_i)Tx\|+\|S_i\|\|(T-T_i)x\|.$$

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