Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP [Formula: see text] that is the image of another GP [Formula: see text] under a linear transformation [Formula: see text] acting on the sample paths of [Formula: see text] are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when [Formula: see text] is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator [Formula: see text] acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille’s theorem for the Bochner integral of a Banach-valued random variable.