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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.

With exact data $f=K{u}^{†}$, the iteration error obeysThe source condition for quadratic regularization $u^{†}=K^{\ast }v$ excludes any null space component. Using its singular system of a compact operator coefficients,By Bernoulli's inequality, $(1+\tau {\sigma }^2{)}^k\ge 1+k\tau {\sigma }^2$. Also $1+k\tau {\sigma }^2\ge 2\sqrt{k\tau }\sigma$. ThereforeSumming the squared coefficients and using Bessel inequality yieldsThis is an a priori error rate from the source condition for quadratic regularization, with constant independent of $k$.

Assume $K{u}^{†}=f$, as in the exact-solution setting used for the preceding source condition for quadratic regularization. Let $r=r_{k+1}$, $\eta =f^{\delta }-f$, and $s=u_{\delta }^{(k+1)}-u_{\delta }^{(k)}=-\tau K^{\ast }r$. Then $K(u_{\delta }^{(k+1)}-u^{†})=r+\eta$, and expanding the two squared errors givesThe Cauchy-Schwarz inequality supplies the last line. Thus while the residual remains at least the noise bound,Compatibility of the exact data is essential. If only $f\in \mathcal{D}(K^{†})$ is retained, let $Kx=(x,0)$, $f=(0,1)$, $f^{\delta }=(1/2,1)$, $\delta =1/2$ and $\tau =1$. The minimum-norm least-squares solution is zero, but the first iterate is $1/4$ and its residual norm is $\sqrt{17}/4>\delta$. Its error has increased from zero. Thus the original assertion uses the implicit exact-data assumption from part (c), and is false without it.