  We consider the family of energy-constrained diamond norms on the set of
Hermitian-preserving linear maps (superoperators) between Banach spaces of
trace class operators. We prove that any norm from this family generates the
strong (pointwise) convergence on the set of all quantum channels (which is
more adequate for describing variations of infinite-dimensional channels than
the diamond norm topology).
  We obtain continuity bounds for information characteristics (in particular,
classical capacities) of energy-constrained quantum channels (as functions of a
channel) with respect to the energy-constrained diamond norms which imply
uniform continuity of these characteristics with respect to the strong
convergence topology.
