Let $A$ be a countable orthonormal set of a Hilbert space $H$. Prove the following.

a) $P_{\overline{\text{span A}}}(x) = \sum_{a\in A}\langle x,a\rangle a \ \text{ for all }x\in H$
b) $\rho(x,\overline{\text{span A}}) = ||x||^2-\sum_{a\in A}|\langle x,a \rangle |^2 \ \ \ \ \forall x\in H$
c)If $\alpha$ is a scalar-valued function on $A$ s.t. $\sum_{a\in A}|\alpha(a)|^2

Question

Let $A$ be a countable orthonormal set of a Hilbert space $H$. Prove the following.

a) $P_{\overline{\text{span A}}}(x) = \sum_{a\in A}\langle x,a\rangle a \ \text{      for all }x\in H$
b) $\rho(x,\overline{\text{span A}}) = ||x||^2-\sum_{a\in A}|\langle x,a \rangle |^2 \  \ \ \ \forall x\in H$
c)If $\alpha$ is a scalar-valued function on $A$ s.t. $\sum_{a\in A}|\alpha(a)|^2<\infty$, then the sum $\sum_{a\in A}\alpha(a)a $ converges

zzr0ck3r · Answer

So there were things I needed to prove before I started this and I have done all of them, namely

if $A:=\{a_n|n\in\mathbb{N}\}$ is an orthonormal set in a Hilbert space $H$, then t.f.a.e. 
(i) $A$ is a orthonormal basis of $H$.
(ii) $\overline{\text{span A}}=H$
(iii) For any $x\in H$, one has $x=\sum_{n=1}^\infty \langle x, a_n \rangle a_n$
(iv) for any $x\in H$, one has $||x||^2=\sum_{n=1}^\infty | \langle x, a_n \rangle |^2$

I also proved $\langle x, a_n \rangle = 0 \ \forall n\in \mathbb{N} \implies \ x=0$

zzr0ck3r · Answer

@eliassaab

zzr0ck3r · Answer

$\rho(x,K)$ is the distance from x to set $K$
$P_k(x)$ is the projection of x onto $K$