= Solution
A <linear regularization> is a family of <bounded linear operators> $R_\alpha:V\to U$, $\alpha>0$, for which $R_\alpha f\to K^\dagger f$ for every $f\in\mathcal D(K^\dagger)$ as $\alpha\downarrow0$. To obtain a <convergent regularization of an inverse problem> for noisy data, choose a <regularization parameter> so that the exact-data approximation error and the amplified noise both vanish.
An example is <Tikhonov regularization>:
$$
\boxed{R_\alpha=(K^*K+\alpha I)^{-1}K^*.}
$$
The positive quadratic term makes $K^*K+\alpha I$ invertible. The <Tikhonov filter norm bound> follows from the filter $s/(s^2+\alpha)$, whose maximum for $s\geq0$ is $1/(2\sqrt\alpha)$. Therefore $\|R_\alpha\|\leq1/(2\sqrt\alpha)$. The spectral factors $s^2/(s^2+\alpha)$ tend to one on the positive spectrum; <dominated convergence theorem> gives consistency on $\mathcal D(K^\dagger)$. The <noise-bias decomposition for linear regularization> then proves convergence whenever $\alpha(\delta)\to0$ and $\delta/\sqrt{\alpha(\delta)}\to0$, for example $\alpha(\delta)=\delta$ for sufficiently small positive noise levels.
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