 Solution

If the $B_k$ form the eigenbasis of $C_X$, thenThe retained functional principal component scores are therefore uncorrelated with every omitted score, so $g=0$. The least-squares estimates of $c_1,\dots ,c_K$ are consistent despite truncation. Fixed $K$ still estimates only the projection ${\beta }_K$; consistency for the complete function requires the truncation level to increase and the tail error to vanish.