  We consider energy-constrained infinite-dimensional quantum channels from a
given system (satisfying a certain condition) to any other systems. We show
that dealing with basic capacities of these channels we may assume (accepting
arbitrarily small error $\varepsilon$) that all channels have the same
finite-dimensional input space -- the subspace corresponding to the
$m(\varepsilon)$ minimal eigenvalues of the input Hamiltonian.
  We also show that for the class of energy-limited channels (mapping
energy-bounded states to energy-bounded states) the above result is valid with
substantially smaller dimension $m(\varepsilon)$.
  The uniform finite-dimensional approximation allows to prove the uniform
continuity of the basic capacities on the set of all quantum channels with
respect to the strong (pointwise) convergence topology. For all the capacities
we obtain continuity bounds depending only on the input energy bound and the
energy-constrained-diamond-norm distance between quantum channels (generating
the strong convergence on the set of quantum channels).
