Let $\mathcal{H}=(H^1(\mathbb{R}^3))^N$ be the product space with the associated norm $$ \Vert U\Vert_1=\left(\sum^N_{i=1}\Vert u_i\Vert_1^2\right)^{1/2} $$ where $U=(u_1,u_2,...,u_N)\in\mathcal{H}$. Let $$ \mathcal{K}=\{U\in\mathcal{H};\int_{\mathbb{R}^3}u_iu_j=\delta_{i,j},\forall 1\leq i,j\leq N\} $$ and $$ \mathcal{H}_n=(X_n)^N\subset\mathcal{H} $$ where $X_n$ is a finite dimensional space of $H^1(\mathbb{R}^3)$.
Define a new projection $\Pi:\mathcal{K}\rightarrow \mathcal{H}_n\cap\mathcal{K}$ as $$ \Vert \Pi U-U\Vert_1=\min\limits_{V\in\mathcal{H}_n\cap\mathcal{K}}\Vert V-U\Vert_1 $$
Is it true that, for $U\in\mathcal{K}$ there exists a constant $C$ independent of $n$ such that $$ \Vert \Pi U-U\Vert_1\leq C\min\limits_{V\in\mathcal{H}_n}\Vert V-U\Vert_1 $$
Thank you very much.
