I am reading the proof of corollary 5.4. in Brezis book "FA, Sobolev Sp. and PDEs", I would like some clarifications.
The statement is the following
Suppose $W$ is a closed linear subspace of $H.$ For $x\in H,$ $y=P_Wx$ is characterized by the property that for all $\omega \in W$ $$ y\in W \ \text{and} \ \langle x-y, \omega \rangle =0.$$
Brezis' proof proceeds like this:
Indeed, if $y=P_Wx$ we have already seen that for all $\omega \in W$ $$\langle x-y,\omega-y \rangle \leq 0,$$ whence for all $t\in \mathbb{R},\omega \in W$ $$\langle x-y,t\omega-y \rangle \leq 0$$
from which we obtain the desired equality $$ y\in W \ \text{and} \ \langle x-y, \omega \rangle =0.$$
(only) question (remaining)
Why this passage? i.e. how does he pass from the inequality for all $t$ to the above equality?
On the other hand, if the equality above holds, we obtain in particular $$\langle x-y,\omega-y \rangle = 0$$ for all $\omega \in W.$
and thus $$y=P_Wx.$$
We have $\langle x-y,tw-y \rangle \leq 0$ for all $t\in\mathbb R$ and $w\in W$. We already have $y\in W$(see Theorem $5.2$ of the same textbook). In particular, for $t=2$ and $w=y$, we get $$\langle x-y,y \rangle \leq 0.$$ Now, rearranging the original inequality, we see that $$t\langle x-y,w \rangle \leq \langle x-y,y \rangle\leq0 $$ for all $t\in\mathbb R$ and $w\in W$. In particular, take $t=\langle x-y,w\rangle\in\mathbb R$(Brezis defines the inner product from $H\times H$ to $\mathbb R$). Then, $\langle x-y,w \rangle^2\leq 0$, from which the conclusion follows.