Approximate source bound for coercive Tikhonov regularization ($\Vert R_{\alpha }f-u^{†}\Vert \le {\eta }_r+\sqrt{\alpha }\beta r$)

Let $K$ be injective, $u^{†}=K^{†}f$, $z={\beta }^{-2}{B}^{\ast }B{u}^{†}$, and ${\eta }_r={\inf }_{\Vert w\Vert \le r}\Vert z-K^{\ast }w\Vert$. Since $\overline{\mathcal{R}(K^{\ast })}=\mathcal{N}(K{)}^{\perp }=U$, ${\eta }_r\to 0$. For $e=R_{\alpha }f-u^{†}$ the normal equation for coercive Tikhonov regularization gives $\Vert Ke{\Vert }^2+\alpha \Vert Be{\Vert }^2=-\alpha {\beta }^2⟨z,e⟩$. Insert an approximate $K^{\ast }w$ and complete the square in $\Vert Ke\Vert$ to obtain $\Vert e{\Vert }^2\le \Vert z-K^{\ast }w\Vert \Vert e\Vert +\alpha {\beta }^2{r}^2/4$. This implies the displayed bound, even with coefficient $1/2$ on its last term, and proves consistency without a fixed exact source condition for quadratic regularization.