# Topologies corresponding to norm, SOT and WOT under duality

Let $X$ be a Banach space with norm dual $X'$. For a linear mapping $T : X \to X$ denote by $T' : X' \to X'$, $(T'f)(x) := f(Tx)$ its adjoint. Denote by $B(X)$ the space of bounded operators on $X$ and by $L_{w^*}(X')$ the space of weakly$^*$-continuous operators on $X'$. Then $T \in B(X)$ iff $T' \in L_{w^*}(X')$. Moreover, any $S \in L_{w^*}(X')$ is of the form $S = T'$ for some $T \in B(X)$ (see here). Hence, we can identify $B(X) = L_{w^*}(X')$ as vector spaces.

On $B(X)$ we can consider the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT). The above identification $B(X) = L_{w^*}(X')$ provides corresponding topologies on $L_{w^*}(X')$.

Now $L_{w^*}(X')$ is a linear subspace of $B(X')$. On $B(X')$ we can also consider these three types of topologies. Are the mentioned topologies on $L_{w^*}(X')$ (coming from $B(X)$) related in some way to these (or other familiar) topologies on $B(X')$? What about the special case of $X = \ell^1$ and $X' = \ell^\infty$?