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.
Serre defines the direct sum (internal) $V=W\oplus W'$, For $x\in V$,the mapping $p(x=w+w')=w$ is the projection map of $V$ onto $ W$ associated with $V=W\oplus W'$; the image of $p$ is $W$ and $p(x)=x$ for all $x\in W$. Conversely, if $p$ is any way satisfying the above two condition; Then we can show that $V$ is the direct sum of $W$ and the kernel of $p$. A bijective correspondence is thus established between the projections of $V$ onto $W$ and the complements of $W$ in $V$.
What is the definition of projection serre uses here?
Suppose if we assume "the mapping $p(x=w+w')=w$ is the projection map of $V$ onto $ W$ associated with $V=W+W'$" as definition.
Then by definition itself we have a bijection between projection and complement, As any projection requires $V$ to be divided into direct sum of $W$ and it's complement. In that case one doesn't need the converse statement he mentioned.
So I'm wondering what is the definition of projection serre assumes here?
[Of course if one use equivalent statement of projection like "the image of $p$ is $W$ and $p(x)=x$" or $p^2=p$. Then one need converse part given by serre.]
I think my question is very trivial but I couldn't figured out
If $W$ is a subspace of a linear space $V$, a projection from $V$ onto $W$ is by definition a linear map $p: V \rightarrow W$ such that $p(x) = x$ for all $x \in W$. A priori this has nothing to do with internal direct sums or complements.
Note that such a map is automatically surjective, and its kernel is a complement to $W$ in $V$, which is to say that $V$ is the internal direct sum of $W$ and $\operatorname{Ker} p$.
In this way we get a mapping
$$p \mapsto \operatorname{Ker} p$$
$$\textrm{projections from $V$ onto $W$} \rightarrow \textrm{complements of $W$ in $V$}$$
Conversely, if $W'$ is a complement to $W$ in $V$, then every $v \in V$ can be uniquely written as $w+w'$ for $w \in W$ and $w' \in W'$, and we can define a projection $p: V \rightarrow W$ from $V$ onto $W$ by $p(v) = w$. In this way we can get a mapping
$$W' \mapsto p$$ $$\textrm{complements of $W$ in $V$} \rightarrow \textrm{projections from $V$ onto $W$}$$
and these two mappings are inverse to one another.