  In recent years, new properties of space-time duality in the Hamiltonian
formalism of certain integrable classical field theories have been discovered
and have led to their reformulation using ideas from covariant Hamiltonian
field theory: in this sense, the covariant nature of their classical $r$-matrix
structure was unraveled. Here, we solve the open question of extending these
results to a whole hierarchy. We choose the Ablowitz-Kaup-Newell-Segur (AKNS)
hierarchy. To do so, we introduce for the first time a Lagrangian multiform for
the entire AKNS hierarchy. We use it to construct explicitly the necessary
objects introduced previously by us: a symplectic multiform, a multi-time
Poisson bracket and a Hamiltonian multiform. Equipped with these, we prove the
following results: $(i)$ the Lax form containing the whole sequence of Lax
matrices of the hierarchy possesses the rational classical $r$-matrix
structure; $(ii)$ The zero curvature equations of the AKNS hierarchy are
multiform Hamilton equations associated to our Hamiltonian multiform and
multi-time Poisson bracket; $(iii)$ The Hamiltonian multiform provides a way to
characterise the infinite set of conservation laws of the hierarchy reminiscent
of the familiar criterion $\{I,H\}=0$ for a first integral $I$.
