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 agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.