Summary: We consider the problem of estimating a signal \(Y\) with values in a Banach space based on the observation \(X\) with values in another Banach space given their joint Gaussian distribution. Linear estimators are defined to be measurable linear transformations. A characterization of measurable linear transformations with respect to a Gaussian measure by radonifying operators is established. The Bayes estimator \(\mathbb{E} (Y |X)\) is shown to be a measurable linear transformation and the associated radonifying operator is derived.