Let $H$ be a Hilbert space and $P: H \rightarrow H$ a linear mapping satisfying $P^2 = P$. Show that
$$F := ranP, P = P_F$$
with
$$ranP := \{Px | x \in H\}$$
Could someone please explain to me what $$P = P_F$$ means? What is $P_F$?
Any hints are more than welcome.