  For a given symmetrically normed ideal I on an infinite dimensional Hilbert
space H, we study the rectifiable distance in the classical Banach-Lie unitary
group $$ U_I={u is a unitary operator in H, u-1\in I}. $$ We prove that
one-parameter subgroups of U_I are short paths, provided the spectrum of the
exponent is bounded by $\pi$, and that any two elements of U_I can be joined
with a short path, thus obtaining a Hopf-Rinow theorem in this infinite
dimensional setting, for a wide and relevant class of (non necessarily smooth)
metrics. Then we prove that the one-parameter groups are the unique short paths
joining given endpoints, provided the symmetric norm considered is strictly
convex.
