$\B_1(\H_E)$. $\square$

Theorem \[KSW-E\] 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 energy-constrained $cb{\hspace{1pt}}\textrm{-}$norm on the set of CP linear maps $\Phi$ and the operator *E*-norm on the set of Stinespring operators $V_{\Phi}$. Its basic assertion is the equality $$\label{ksw-rel-+}
\beta_E^G(\Phi,\Psi)=\inf_{V_{\Phi},V_{\Psi}} \|V_{\Phi}-V_{\Psi}\|_E^G.$$ Some difficulty of applying Theorem \[KSW-E\] is related to the fact that the infimum in (\[ksw-rel-+\]) is over all common Stinespring representation (\[c-S-r\]). But by using the constructions from the proof of this theorem one can obtain its versions which are more convenient for applications, in particular, for analysis of converging sequences of CP linear maps.

\[KSW-E+\] *Let ${G}$ be a positive semidefinite densely defined operator on $\H_A$ satisfying condition (\[H-cond\]), $\beta_E^G$ and $\|\!\cdot\!\|_E^G$ be, respectively, the energy-constrained Bures distance and the operator $E\textrm{-}$norm induced by ${G}$. Let $\,\Phi$ be a CP linear map from $\T(\H_A)$ to $\T(\H_B)$.*

A\) *There is a Stinespring representation of $\,\Phi$ with the operator $\,V'_{\Phi}:\H_A\rightarrow\H_B\otimes\H_{E'}$ such that $$\label{ksw-rel+}
\beta_E^G(\Phi,\Psi)=\inf_{V_{\Psi}}
 \|V'_{\Phi}-V_{\Psi}\|_E^G,$$ for any CP linear map $\Psi:\T(\H_A)\rightarrow\T(\H_B)$, where the infimum is over all Stinespring representations of $\,\Psi$ with the same environment space $\,\H_{E'}$. The infimum in (\[ksw-rel+\]) is attainable.*

B\) *If $\,V_{\Phi}:\H_A\rightarrow\H_B\otimes\H_{E}$ is 