Does someone know if in the problem the projection of x onto U is defined like that :
$x_u = \displaystyle \frac{\langle x,u\rangle}{u. u}$ $u$
Problem:
Let $U,V\subset\mathbb C^n$ be two subspaces, such that $\mathbb C^n = U+V$ and further assume $U\cap V = \{0\}$.
Show that every $x\in\mathbb C^n$ can be written as $x=x_u+x_v$ with $x_u\in U$ and $x_v\in V$ and that this decomposition is unique.
If $x\in \mathbb{C}^n$, then since $\mathbb{C}^n = U+V$ there must exist $x_u\in U$ and $x_v\in v$ such that $$x=x_u+x_v$$ Suppose now that there exist alternative $y_u,y_v$ with $$x=y_u+y_v$$ Then $$x-x=0=(x_u-y_u)+(x_v-y_v)$$ Hence $$y_u-x_u=x_v-y_v$$ Since $U$ and $V$ are subspaces, $y_u-x_u\in U$ and $x_v-y_v\in V$. Thus these vectors are in $U\cap V=\{0\}$, hence $y_u-x_u=x_v-y_v=0$, and hence the decomposition is unique.
Clearly we didn't need projection here. Indeed this proof works for infinite dimensional vector spaces as well, as well as vector spaces over an arbitrary field (not necessarily real or complex numbers, or even characteristic $0$), and an inner product is not required.