  The Koopman representation is an infinite dimensional linear representation
of linear or nonlinear dynamical systems. It represents the dynamics of output
maps (aka observables), which are functions on the state space whose evaluation
is interpreted as an output. Conceptually simple derivations and commentary on
the Koopman representation are given. We emphasize an important duality between
initial conditions and output maps of the original system, and those of the
Koopman representation. This duality is an important consideration when this
representation is used in data-driven applications such as the Dynamic Mode
Decomposition (DMD) and its variants. The adjoint relation between the Koopman
representation and the transfer operator of mass transport is also shown.
