So I'm reading through Serre's "Linear Representations of Finite Groups," and I'm a bit confused by what's probably a fairly minor point. However, subsequent proofs are hinging on it, so I figure I'll turn to you guys for clarification.
So he defines the direct sum (I guess in this case the internal one), and then introduces the projection map. So if $V=W+W'$, then $p(x=w+w')=w$. Then, for some converse implicating, he assumes we have some map $p$ from $V$ to $V$ whose image is $W$ and whose restriction to $W$ yields $p(x)=x$. Then we can show that $V$ is the direct sum of $W$ and the kernel of $p$. It's the next line that trips me up. He says "A bijective correspondence is thus established between the projections of $V$ onto $W$ and the complements of $W$ in $V$." My brain can't fill the gap in the logic, nor can I really understand what that statement is all about.
Apologies for the logorrhea. Any and all help is very much appreciated!