The KSW theorem shows continuity of the map $V_{\Phi}\mapsto\Phi$ and selective continuity of the multi-valued map $\Phi\mapsto V_{\Phi}$ with respect to the $cb{\hspace{1pt}}$-norm topology on the set $\F(A,B)$ of all CP linear maps $\Phi$ from $\T(\H_A)$ to $\T(\H_B)$ and the operator norm topology on the set of Stinespring operators $V_{\Phi}$.

The $cb{\hspace{1pt}}$-norm topology is widely used in the quantum theory, by it is too strong for description of physical perturbations of infinite-dimensional quantum channels [@SCT; @W-EBN]. Our aim is to obtain a version of the KSW theorem which would show continuity of the map $V_{\Phi}\mapsto\Phi$ and selective continuity of the multi-valued map $\Phi\mapsto V_{\Phi}$ with respect to *weaker topologies* on the sets of CP linear maps $\Phi$ and Stinespring operators $V_{\Phi}$. A natural way to do this is to use the operator *E*-norms induced by some positive operator ${G}$ on $\H_A$ (naturally generalized to operators between different separable Hilbert spaces, see Remark \[en-gen\]) and the energy-constrained $cb{\hspace{1pt}}$-norms $$\label{E-sn}
  \|\Phi\|^G_{\mathrm{cb},E}\doteq\sup_{\rho\in\S(\H_{AR}):{\mathrm{Tr}}{G}\rho_A\leq E}\|\Phi\otimes {\mathrm{Id}}_R(\rho)\|_1,\quad E>0,\quad (\textrm{where}\; \rho_A\doteq{\mathrm{Tr}}_R{\hspace{1pt}}\rho)$$ on the set of Hermitian-preserving linear maps from $\T(\H_{A})$ to $\T(\H_{B})$ introduced independently in [@SCT] and [@W-EBN] (the positive operator ${G}$ is treated therein as a Hamiltonian of a quantum system $A$).[^13] If ${G}$ is a 