I am trying to understand the proof of the existence of orthogonal projections. The statement is given as follow: Let $V$ be an Hilbert space and $W \subseteq V$ be a closed subset, then $\exists$! linear and cont. function $\pi: V \to V$ s.t. $\forall x \in V$,
$\pi(x) \in W$
$x - \pi (x) \in W^{\bot}$
The proof first show the existence and the uniqueness of $\pi$ but then I got stucked in the proof of the linearity... Let $x,y \in V$ and $\alpha, \beta \in \Bbb R$ so that $\alpha \pi (x) + \beta \pi (y) \in W$. But now we have that $$\alpha x + \beta y -(\alpha \pi (x) + \beta \pi (y))= \alpha (x - \pi (x)) + \beta (y- \pi(y)) \in W ^{\bot}$$ By uniqueness of $\pi$ we follow that $$\pi(\alpha x + \beta y) = \alpha \pi (x) + \beta \pi (y)$$ and hence that $\pi$ is linear.
I don't understand here the fact that we used the uniqueness to conclude the linearity... How does we can suddenly apply $\pi$ to $\alpha x + \beta y$? Many thanks for some helps...
What the first part of the proof probably shows is that for each vector $v$, there exists a unique vector $\pi(v)$ satisfying (1) and (2). Now taking $v = \alpha x + \beta y$, the vector $\alpha \pi(x) + \beta \pi(y)$ also satisfies (1) and (2). But $\pi(v)$ is the unique vector with this property. So we must have $\pi(v) = \alpha \pi(x) + \beta \pi(y)$.