Independent approximation in separable Hilbert spaces via spectral truncation

This note studies independent approximation in separable Hilbert spaces relative to a given sub-σ-algebra. For the residual after conditioning, we consider finite-dimensional spectral truncations associated with its covariance operator and combine them with a decomposition result in Euclidean spaces. We show that each truncation admits a representation by sums of random terms independent of the prescribed information set. Moreover, these truncations are optimal in mean square within the natural class of finite-dimensional approximations generated by such independent sums. As a consequence, the best approximation error is exactly determined by the spectral tail of the covariance operator.