Orthogonal projection in $L^2(\Omega)$ and $W_{0}^{1,2}(\Omega)$ While stydying the proof of the existence theorem for weak solutions for parabolic equations using the Galerkin approximation I encountered the following problem:
Assume that $\Omega \subseteq {\mathbb{R}}^{d}$ is an open set and ${\left\lbrace w_k \right\rbrace}_{k=1}^{\infty}$ is an orthonormal basis of $L^2(\Omega)$ such that ${\left\lbrace w_k \right\rbrace}_{k=1}^{\infty}$ is also orthogonal in $W_{0}^{1,2}(\Omega)$. For every $n \in \mathbb{N}$ let $P_n$ be the $L^2$-orthogonal projection onto $\mathrm{span} {\left\lbrace w_k \right\rbrace}_{k=1}^{n}$, i.e.
$$P_n (u)=\sum\limits_{k=1}^{n} {\left( u , w_k \right)}_{L^2(\Omega)}w_k = \sum\limits_{k=1}^{n} \left( \int\limits_{\Omega}u(x)w_k(x) \mathrm{d}x \right)w_k \ , \ \ \ \ \ \ u \in L^2(\Omega).$$
It is clear that $\| P_n (u) \|_{L^2(\Omega)} \leq \|u\|_{L^2(\Omega)}$ for every $n \in \mathbb{N}$ and $u \in L^2(\Omega)$. However, what I need is the following:
$$\exists \, C>0 \ \ \forall \, n \in \mathbb{N} \ \ \forall \, u \in W_{0}^{1,2}(\Omega):\| P_n (u) \|_{W_{0}^{1,2}(\Omega)} \leq C \|u\|_{W_{0}^{1,2}(\Omega)}.$$
I'm not even sure it is true, but I need it to obtain some a priori estimates.
I'll appreciate any help.
 A: Note: Below I choose a specific basis (the eigenfunctions of the inverse Laplacian). I am not sure if it works for any ONB of $L^2$, as one is doing it in finite elements.
Let us denote $H=L^2(\Omega)$ and $V=H_0^1(\Omega)$.
Since $\{w_k\}_k$ is an orthonormal basis in $H$, we can write the orthogonal projection from $H$ to $\text{span}\{w_1,...,w_m\}$ via
$$P_m h = \sum_{i=1}^m (h,w_i)_H w_i \text{ for all } h \in H.$$  On the one hand we have for all $v \in V$
$$\|P_m v\|_V^2=(P_m v,P_m v)_V=\sum_{i,j=1}^m (v,w_i)_H (v,w_j)_H \underbrace{(w_i,w_j)_V}_{=\delta_{i,j}/\lambda_i}=\sum_{i=1}^m \frac{1}{\lambda_i} |(v,w_i)_H|^2,$$
where $\lambda_i$ is the eigenvalue to the eigenfunction $w_i$ of $A^{-1}J$ (here $A:V \to V'$ is the Stokes operator and $J:V \to H$ is the injection operator), but on the other hand we have for all $v \in V$
$$\begin{aligned}\|v\|_V^2 =(v,v)_V &=\lim_{n \to \infty} \left( \sum_{i=1}^n (v,w_i)_H w_i, \sum_{j=1}^n (v,w_j)_H w_j \right)_V \\ &=\lim_{n \to \infty} \sum_{i,j=1}^n (v,w_i)_H (v,w_j)_H \underbrace{(w_i,w_j)_V}_{=\delta_{i,j}/\lambda_i} \\ &=\sum_{i=1}^\infty \frac{1}{\lambda_i} |(v,w_i)_H|^2. \end{aligned}$$
Thus, comparing these two equations yields
\begin{equation} 
 \|P_m v\|_V \leq \|v \|_V \text{ for all } v \in V. \label{EqProjIneq}
\end{equation}
