Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by $$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to 0 ).$$

We can identify $M(A) \cong \mathcal{L}(A)$ where $\mathcal{L}(A)$ are the adjoinable operators of the right Hilbert $A$-module $A$ with respect to the inner product $\langle a,b \rangle:= a^*b.$ The strict topology on $\mathcal{L}(A)$ is given as follows:

$$t_\lambda \to t \iff \forall a \in A: (\|t_i(a)-t(a)\| + \|t_i^*(a) -t^*(a)\| \to 0).$$

When we identify $A$ as a subset of $\mathcal{L}(A)$ via the mapping $ab^* \mapsto \theta_{a,b}$ where $\theta_{a,b}(x) = a\langle b,x\rangle$, we obtain two notions of strict topologies on $M(A)$. I guess on bounded sets these topologies agree, but are these topologies in general equal?