I'm trying to work through this proof in Serre:
http://www.math.ualberta.ca/~schlitt/serretheorem.png
I'm confused how $W^{0}$ is different from $W$.
If I have a subspace $W$ of a vector space $V$, and a projection $p:V\to W$, I get a complement $W^{c}$ of $W$ such that $V = W^{c}\oplus W$. Intuitively I would just take $W^{c}$ to be the span of all basis vectors of $V$ not contained in $W$.
So in the proof we construct a new projection $p^{0}$, from $V$ onto $W$ given an arbitrary starting projection $p$, but I don't see why $W'$ ( the complement we start with ) would be any different than the one we end up with $W^{0}$. Can someone help me understand how complements are constructed? I think it's assumed as basic background understanding in the proof.
I think I may have clued into a partial answer, so the complement is just the kernel of the projection?