Strong maximum a posteriori estimation in Banach spaces with Gaussian priors

We show that a nonparametric Bayesian inverse problem with a Gaussian prior has a strong maximum a posteriori estimator in the sense of small-ball modes as introduced by Dashti et al. (2013). This result holds in any separable Banach space under mild conditions on the forward operator and extends existing results by significantly relaxing the conditions on the log-likelihood and on the space in which the inverse problem is set.