  An asymptotic theory is established for linear functionals of the predictive
function given by kernel ridge regression, when the reproducing kernel Hilbert
space is equivalent to a Sobolev space. The theory covers a wide variety of
linear functionals, including point evaluations, evaluation of derivatives,
$L_2$ inner products, etc. We establish the upper and lower bounds of the
estimates and their asymptotic normality. It is shown that $\lambda\sim n^{-1}$
is the universal optimal order of magnitude for the smoothing parameter to
balance the variance and the worst-case bias. The theory also implies that the
optimal $L_\infty$ error of kernel ridge regression can be attained under the
optimal smoothing parameter $\lambda\sim n^{-1}\log n$. These optimal rates for
the smoothing parameter differ from the known optimal rate $\lambda\sim
n^{-\frac{2m}{2m+d}}$ that minimizes the $L_2$ error of the kernel ridge
regression.
