Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 36 / 1 / d / Solution

The map $h\mapsto Ph$ is a bounded linear functional on L2 space, since $|Ph|\le \Vert h{\Vert }_2$ by the Cauchy-Schwarz inequality. Its kernel, the mean-zero L2 space, is therefore closed. Every nuisance score function and the parametric score function $\dot{\ell }$ are centered, so both ${\Pi }_{\mathcal{N}}\dot{\ell }$ and $\tilde{\ell }$ are centered. HenceMoreover, the efficient score belongs to the orthogonal complement of the nuisance tangent space. Writing $\dot{\ell }=\tilde{\ell }+{\Pi }_{\mathcal{N}}\dot{\ell }$ givesThis is the efficient-score projection identity; it remains valid when the efficient information is zero.