= Approximate source bound for coercive Tikhonov regularization
{title2=$\|R_\alpha f-u^\dagger\|\leq\eta_r+\sqrt\alpha\beta r$}
Let $K$ be injective, $u^\dagger=K^\dagger f$, $z=\beta^{-2}B^*Bu^\dagger$, and $\eta_r=\inf_{\|w\|\leq r}\|z-K^*w\|$. Since $\overline{\mathcal R(K^*)}=\mathcal N(K)^\perp=U$, $\eta_r\to0$. For $e=R_\alpha f-u^\dagger$ the <normal equation for coercive Tikhonov regularization> gives $\|Ke\|^2+\alpha\|Be\|^2=-\alpha\beta^2\langle z,e\rangle$. Insert an approximate $K^*w$ and complete the square in $\|Ke\|$ to obtain $\|e\|^2\leq\|z-K^*w\|\|e\|+\alpha\beta^2r^2/4$. 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>.
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