  We consider the statistical inverse problem of recovering an unknown function
$f$ from a linear measurement corrupted by additive Gaussian white noise. We
employ a nonparametric Bayesian approach with standard Gaussian priors, for
which the posterior-based reconstruction of $f$ corresponds to a Tikhonov
regulariser $\bar f$ with a reproducing kernel Hilbert space norm penalty. We
prove a semiparametric Bernstein-von Mises theorem for a large collection of
linear functionals of $f$, implying that semiparametric posterior estimation
and uncertainty quantification are valid and optimal from a frequentist point
of view. The result is applied to study three concrete examples that cover both
the mildly and severely ill-posed cases: specifically, an elliptic inverse
problem, an elliptic boundary value problem and the heat equation. For the
elliptic boundary value problem, we also obtain a nonparametric version of the
theorem that entails the convergence of the posterior distribution to a
prior-independent infinite-dimensional Gaussian probability measure with
minimal covariance. As a consequence, it follows that the Tikhonov regulariser
$\bar f$ is an efficient estimator of $f$, and we derive frequentist guarantees
for certain credible balls centred at $\bar{f}$.
