Projection and quadrature errors split in L2

For each $\beta \in \mathrm{\Lambda }$, the coefficient formula [F2], orthonormality [F3], and bilinearity [F4] give $⟨u-P_{\mathrm{\Lambda }}u,{\psi }_{\beta }⟩=c_{\beta }-{\sum }_{\alpha \in \mathrm{\Lambda }}{c}_{\alpha }{\delta }_{\alpha \beta }=0$. Thus $u-P_{\mathrm{\Lambda }}u$ is orthogonal to the finite span ${\mathcal{P}}_{\mathrm{\Lambda }}$, including when $\mathrm{\Lambda }$ is empty. Subtracting the two finite expansions gives ${\widehat{P}}_{\mathrm{\Lambda }}u-P_{\mathrm{\Lambda }}u={\sum }_{\alpha \in \mathrm{\Lambda }}({\widehat{c}}_{\alpha }-c_{\alpha }){\psi }_{\alpha }\in {\mathcal{P}}_{\mathrm{\Lambda }}$, so this difference is orthogonal to the residual.

The identity $u-{\widehat{P}}_{\mathrm{\Lambda }}u=(u-P_{\mathrm{\Lambda }}u)-({\widehat{P}}_{\mathrm{\Lambda }}u-P_{\mathrm{\Lambda }}u)$ is an orthogonal sum by step 1.1. Expanding its squared norm using [F4] gives $\mathrm{\parallel }u-{\widehat{P}}_{\mathrm{\Lambda }}u{\mathrm{\parallel }}^2=\mathrm{\parallel }u-P_{\mathrm{\Lambda }}u{\mathrm{\parallel }}^2+\mathrm{\parallel }{\widehat{P}}_{\mathrm{\Lambda }}u-P_{\mathrm{\Lambda }}u{\mathrm{\parallel }}^2$; orthonormality [F3] identifies the second term with ${\sum }_{\alpha \in \mathrm{\Lambda }}\mathrm{\mid }{\widehat{c}}_{\alpha }-c_{\alpha }{\mathrm{\mid }}^2$. To obtain the final inequality, use [F5] to expand $\mathrm{\parallel }v+w{\mathrm{\parallel }}_2^2$ and deduce $\mathrm{\parallel }v+w{\mathrm{\parallel }}_2\le \mathrm{\parallel }v{\mathrm{\parallel }}_2+\mathrm{\parallel }w{\mathrm{\parallel }}_2$ for $v=u-P_{\mathrm{\Lambda }}u$ and $w=P_{\mathrm{\Lambda }}u-{\widehat{P}}_{\mathrm{\Lambda }}u$.