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Let $A\subseteq H$ be a closed subspace of a Hilbert space $H$. Show that for each $f\in H$, there exists a unique element $a\in A$ so that $(a - f)\perp A$ (i.e. $a-f\in A^\perp$).

By definition, a Hilbert space is a complete inner product space. Since $A$ is closed in $H$, it is also complete. Also, one can choose $y_n$ so that $\lim\limits_{n\to\infty} \lVert f - y_n\rVert = \mathrm{dist} (f,A) := \inf\{\lVert f- a\rVert : a \in A\}.$ Then one could let $a = \lim\limits_n y_n,$ assuming it exists (to show it's in $A$ it suffices to show that the sequence is Cauchy). Why does it exist?

If this holds, I think I can show that $b\in A, \langle a - f,b\rangle = 0,$ but I'm not sure how.

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2 Answers 2

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Step 1. Denote $d=\mathrm{dist}(f,A) := \inf\{\lVert f- a\rVert : a \in A\}.$ For each $n\in\mathbb N$, we can take $y_n\in A$ such that $\|f-y_n\|<d+\frac1n$. We show that $\{y_n\}$ is Cauchy.

We use Parallelogram identity to compute $\|y_n-y_m\|^2$. For each $n,m\in\mathbb N$, we have \begin{align*} \|y_n-y_m\|^2&=\|(f-y_n)-(f-y_m)\|^2\\ &=2\left(\|f-y_n\|^2+\|f-y_m\|^2\right)-4\left\|f-\frac{y_n+y_m}2\right\|^2\\ &\leq 2\left(\left(d+\frac1n\right)^2+\left(d+\frac1m\right)^2\right)-4d^2\\ &=4\left(\frac1n+\frac1m\right)d+2\left(\frac1{n^2}+\frac1{m^2}\right). \end{align*} From this we can easily show that $\{y_n\}$ is Cauchy.

Since $A$ is closed, the limit $a=\lim y_n$ belongs to $A$. We also have $\|f-a\|=d$.

Step 2. Now we prove that $\langle a - f,b\rangle = 0$ for any $b\in A$. We may assume WLOG that $b\neq 0$. For any $t\in\mathbb R$, we have $tb+a\in A$, hence $\|a-f+tb\|^2\geq d^2=\|a-f\|^2$ for all $t$, i.e., (Here I assume that the Hilbert space is over $\mathbb R$. If $H$ is a complex Hilbert space, this argument needs to be slightly modified.) $$t^2\|b\|^2+2\langle a - f,b\rangle t+\|a-f\|^2\geq\|a-f\|^2,\qquad\forall t\in\mathbb R.$$ Or equivalently, $$t^2\|b\|^2+2\langle a - f,b\rangle t\geq 0,\qquad\forall t\in\mathbb R.$$ This implies that $\langle a - f,b\rangle = 0$ by taking $t=-\frac{\langle a-f,b\rangle}{\|b\|^2}$. Indeed, this $t$ also can be used for complex Hilbert space $H$! If $H$ is complex and $t=-\frac{\langle a-f,b\rangle}{\|b\|^2}\in\mathbb C$, then we have \begin{align*} \|a-f\|^2\leq \|a-f+tb\|^2&=\|a-f\|^2+\bar t\langle a-f,b\rangle+t\overline{\langle a-f,b\rangle}+|t|^2\|b\|^2\\ &=\|a-f\|^2-\frac{|\langle a-f,b\rangle|^2}{\|b\|^2}, \end{align*} which gives that $\langle a-f,b\rangle=0$.

Step 3. Uniqueness. If $a, a_1\in A$ satisfy $(f-a)\perp A$ and $(f-a_1)\perp A$, then $a-a_1=(f-a_1)-(f-a)\in A^\perp$. That is, $\langle a-a_1, b\rangle =0$ for all $b\in A$. Taking $b=a-a_1\in A$ gives that $\|a-a_1\|=0$ and thus the uniqueness.

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  • $\begingroup$ Thanks. Could you also prove that $a$ is unique? Is that because for any other $c\neq a\in A,$ we have $\lVert f-c\rVert^2 = \lVert a - c \rVert^2 + \lVert f - a\rVert^2 > \lVert f -a\Rvert^2$? $\endgroup$
    – Gord452
    Aug 6 at 19:23
  • $\begingroup$ Also for the complex case, I think the following works: choose $t = -\dfrac{\langle c,b\rangle }{\lVert b\rVert^2}, c = a-f$. Then $\lVert a - f +tb\rVert^2 = \lVert a-f\rVert^2 - \dfrac{|\langle c,b\rangle|^2}{\lVert b\rVert^2} \ge \lVert a - f +tb\rVert^2 \Rightarrow \langle c,b\rangle = 0.$ $\endgroup$
    – Gord452
    Aug 6 at 19:29
  • $\begingroup$ @Gord452 Sorry. I forgot to prove the uniqueness. Your argument of uniqueness proves the uniqueness of $a$ satisfying $\|f-a\|=d$, which also implies the uniqueness of $a$ satisfying $f-a\in A^\perp$. It is fine. But we can directly prove the uniqueness of $a$ satisfying $f-a\in A^\perp$. See my Step 3. Your argument regarding the complex case is great! I've added it to my answer. My previous thought on this was more complicated than yours. Thank you for sharing your ideas! Cheers :) $\endgroup$
    – Feng
    Aug 7 at 2:05
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The parallelogram law is useful in this regard: $$ \|x-y\|^2+\|x+y\|^2 = 2\|x\|^2+2\|y\|^2. $$ For $z\notin A$, let $\{ x_n \}_{n=1}^{\infty}$ be a sequence of vectors in $A$ chosen so that $$ \|z-x_n\| \le \mbox{dist}(z,A)+\frac{1}{n} \tag{*} $$ Applying the parallelogram law to $x=(z-x_n),y=(z-x_m)$ gives $$ \|x_n-x_m\|^2+4\|z-\frac{1}{2}(x_n+x_m)\|^2 =2\|z-x_n\|^2+2\|z-x_m\|^2 $$ Therefore, $$ \|x_n-x_m\|^2+4\cdot\mbox{dist}(z,A)^2 \\ \le 2\left(\mbox{dist}(z,A)+\frac{1}{n}\right)^2+2\left(\mbox{dist}(z,A)+\frac{1}{m}\right)^2 \\ = 4\cdot\mbox{dist}(z,A)^2+4\left(\frac{1}{n}+\frac{1}{m}\right)\mbox{dist}(z,A)+2\left(\frac{1}{n^2}+\frac{1}{m^2}\right) $$ This gives $$ \|x_n-x_m\|^2 \le 4\left(\frac{1}{n}+\frac{1}{m}\right)\mbox{dist}(z,A)+2\left(\frac{1}{n^2}+\frac{1}{m^2}\right). $$ Therefore $\{ x_n \}_{n=1}^{\infty}$ is a Cauchy sequence of vectors in $A$, which must converge to some $a\in A$ because $\mathcal{H}$ is complete and $A$ is closed. Furthermore $\|z-a\| \le \mbox{dist}(z,A) \le \|z-a\|$, which, by equation $(*)$ gives $$ \|z-a\|=\mbox{dist}(z,A). $$ Therefore, there exists a closest point projection of $z$ onto $A$. Furthermore $a\in A$ is unique in this regard because, if $a'\in A$ satisfies $\|z-a\|=\|z-a'\|=\mbox{dist}(z,A)$, then $a=a'$ because of the parallelogram law; indeed $$ \|(z-a)+(z-a')\|^2+\|(z-a)-(z-a')\|^2 \\ = 2\|z-a\|^2+2\|z-a'\|^2 $$ implies $$ 4\|z-\frac{1}{2}(a+a')\|^2+\|a-a'\|^2=4\mbox{dist}(z,A)^2 \\ 4\mbox{dist}(z,A)^2+\|a-a'\|^2 \le 4\mbox{dist}(z,A)^2 \\ \implies a=a'. $$ Therefore, all closest point projections to $A$ are the same. I'll leave it to you to show that a closest point projection $x_p$ of $x$ onto $A$ is the orthogonal projection of $x$ onto $A$.

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