  We present a systematic study of conditional triangular transport maps in
function spaces from the perspective of optimal transportation and with a view
towards amortized Bayesian inference. More specifically, we develop a theory of
constrained optimal transport problems that describe block-triangular Monge
maps that characterize conditional measures along with their Kantorovich
relaxations. This generalizes the theory of optimal triangular transport to
separable infinite-dimensional function spaces with general cost functions. We
further tailor our results to the case of Bayesian inference problems and
obtain regularity estimates on the conditioning maps from the prior to the
posterior. Finally, we present numerical experiments that demonstrate the
computational applicability of our theoretical results for amortized and
likelihood-free inference of functional parameters.
