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.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.