  The linear conditional expectation (LCE) provides a best linear (or rather,
affine) estimate of the conditional expectation and hence plays an important
r\^ole in approximate Bayesian inference, especially the Bayes linear approach.
This article establishes the analytical properties of the LCE in an
infinite-dimensional Hilbert space context. In addition, working in the space
of affine Hilbert--Schmidt operators, we establish a regularisation procedure
for this LCE. As an important application, we obtain a simple alternative
derivation and intuitive justification of the conditional mean embedding
formula, a concept widely used in machine learning to perform the conditioning
of random variables by embedding them into reproducing kernel Hilbert spaces.
