We present a denotational semantics for higher-order probabilistic programs in terms of linear operators between Banach spaces. Our semantics is rooted in the classical theory of Banach spaces and their tensor products, but bears similarities with the well-known semantics of higher-order programs a la Scott through the use of ordered Banach spaces which allow definitions in terms of fixed points. Our semantics is a model of intuitionistic linear logic: it is linear because it is based on a symmetric monoidal closed category of ordered Banach spaces, but by constructing an exponential comonad we can also accommodate non-linear reasoning. We apply our semantics to the verification of the classical Gibbs sampling algorithm.