What is the formula of an orthogonal projection $P$ onto a subspace $V$? Let $H$ be a Hilbert space and let $V$ be a closed subspace in $H$.
I try to find the formula of the orthogonal projection $P$ onto the subspace $V$ ?
An other question.
What is the formula of the orthogonal projection $P$ if the closed supspace $V$ is the image of an other projection $Q$ ($Q$ not required to be orthogonal) ?
Thank you.
 A: You need to have a Hilbert basis of $V$, $(e_i)_{i\in I}$ for that. The orthogonal projection of a vector $x\in H$ onto $V$ is given by the formula:
$$P(x)=\sum_{i\in I}\langle x, e_i\rangle e_i.$$
A: The orthogonal projection $P_Vx$ of $x$ onto the closed subspace $V$ is the unique $v\in V$ such that $(x-v)\perp V$. Such a $v$ exists if $V$ is a complete subspace of an inner product space and, hence, also, if $V$ is a closed subspace of Hilbert space.
Orthogonal projection and closest point projection are the same in this context. That is, if $V$ is a subspace of an inner product space, then $v \in V$ is the orthogonal projection of $x$ onto $V$ iff $v$ is the closest point in $V$ to $x$.
If $V$ is a complete subspace of an inner product space, and if $\{ x_{\alpha} \}_{\alpha\in\Lambda}$ is a complete orthonormal basis of $V$, then
$$
           P_{V}x = \sum_{\alpha\in\Lambda}\langle x,e_{\alpha}\rangle e_{\alpha}.
$$
A: By definition $P: H \rightarrow V$ is a projection of $H$ onto $V$ if $PH \subset V$ and $P^2 = P$. 
Hint 1: if $H$ is separable then so too is any subspace of $H$. If $x_1,\dots,x_n,\dots$ is an orthogonal basis for $H$ then there is some subsequence $x_{s_1},x_{s_2},\dots,x_{s_n},\dots$ of this basis which is a basis of $V$. What does $Px \in V$ mean when we write $x = \sum_{i}a_ix_i$ for $x \in V$? What needs to happened to basis elements not in the basis of $V$?
Hint 2: If $V = Im(Q)$ for some projection $Q$, then $Pq = PQx$ for some $x$ in $H$ whenever $q \in V$. What are the possibilities for $PQ$ given the result of Hint 1? 
