  Relations between states and maps, which are known for quantum systems in
finite-dimensional Hilbert spaces, are formulated rigorously in geometrical
terms with no use of coordinate (matrix) interpretation. In a tensor product
realization they are represented simply by a permutation of factors. This leads
to natural generalizations for infinite-dimensional Hilbert spaces and a simple
proof of a generalized Choi Theorem. The natural framework is based on spaces
of Hilbert-Schmidt operators $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$ and
the corresponding tensor products $\mathcal{H}_1\otimes\mathcal{H}_2^*$ of
Hilbert spaces. It is proved that the corresponding isomorphisms cannot be
naturally extended to compact (or bounded) operators, nor reduced to the
trace-class operators. On the other hand, it is proven that there is a natural
continuous map
$\mathcal{C}:\mathcal{L}_1(\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1))\to
\mathcal{L}_\infty(\mathcal{L}(\mathcal{H}_2),\mathcal{L}_1(\mathcal{H}_1))$
from trace-class operators on $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$
(with the nuclear norm) into compact operators mapping the space of all bounded
operators on $\mathcal{H}_2$ into trace class operators on $\mathcal{H}_1$
(with the operator-norm). Also in the infinite-dimensional context, the Schmidt
measure of entanglement and multipartite generalizations of state-maps
relations are considered in the paper.
