In this work we extend the scope of the classical Cram´er-Rao lower bound, or information inequality, from Euclidean to function spaces. In other words we derive a tight lower bound on the autocovariance function of a function estimator. We do this in the context of system identification. Two key elements of system identification are experiment design and model selection. The novel information inequality on function spaces is important for model selection because it allows the user to compare estimators using different model structures. We provide a consistent treatment of the case where the Fisher information matrix is singular. This makes it possible to take into account that in optimal experiment design one tries to mask those parts of the system non-identifiable, which are irrelevant for the application.