Mean-zero L2 space ($L_0^2(P)=\{h\in L^2(P):Ph=0\}$)

The mean-zero L2 space is the kernel of the bounded linear functional $h\mapsto Ph$ on L2 space for a probability measure $P$. It is a closed subspace of a Hilbert space, with inner product $P(hk)$, and is the natural ambient space for score functions and canonical gradients.