I'm doing Problem II.3.12 in textbook Analysis I by Amann/Escher.
Could you please verify whether my attempt on (b) is fine or contains logical gaps/errors? Any suggestion is greatly appreciated!
My attempt:
We have $x-p_{F}(x) \in F^{\perp}$ from (a) and $p_{F}(x)-y \in F$ for all $y \in F$. As such, $x-p_{F}(x) \perp p_{F}(x)-y$ and so $\| (x-p_{F}(x)) + (p_{F}(x)-y) \|^{2} = \|x-p_{F}(x)\|^2 +$ $\|p_{F}(x)-y\|^{2}$. It follows from $\|p_{F}(x)-y\|^{2} = 0$ when $y = p_{F}(x)$ that $\inf_{y \in F} \|p_{F}(x)-y\|^{2}= 0$.
We have $$\begin{aligned} \|x-y\|^{2} &= \| (x-p_{F}(x)) + (p_{F}(x)-y) \|^{2} \\ &= \| x-p_{F}(x)\|^2 + \|p_{F}(x)-y\|^{2} \end{aligned}$$
Then $$\begin{aligned} \left( \inf_{y \in F} \|x-y\| \right)^2 &= \inf_{y \in F} \|x-y\|^{2} \\
&= \inf_{y \in F} \left (\|x-p_{F}(x)\|^2 + \|p_{F}(x)-y\|^{2} \right) \\
&= \inf_{y \in F} \|x-p_{F}(x)\|^2 + \inf_{y \in F} \|p_{F}(x)-y\|^{2} \\
&= \|x-p_{F}(x)\|^2 + 0\end{aligned}$$
Thus $\|x-p_{F}(x)\| = \inf_{y \in F} \|x-y\|$.
Best Answer
This is almost fine. In the series of equations, what you actually have is that $$\inf_{y \in F} \left (\|x-p_{F}(x)\|^2 + \|p_{F}(x)-y\|^{2} \right) \geq\inf_{y \in F} \|x-p_{F}(x)\|^2 + \inf_{y \in F} \|p_{F}(x)-y\|^{2}$$ So what you obtain is $$\inf_{y\in F}\|x-y\|\geq \|x-p_F(x)\|.$$ But since $p_F(x)\in F$, the reverse inequality also holds, so equality holds.